mirror of
https://github.com/vale981/arb
synced 2025-03-04 17:01:40 -05:00
more hypergeometric helper functions
This commit is contained in:
Fredrik Johansson
parent
9a0ad0562c
commit
030a37acb9
8 changed files with 430 additions and 94 deletions
|
|
@ -35,6 +35,15 @@ extern "C" {
|
|||
void acb_hypgeom_pfq_bound_factor(mag_t C,
|
||||
acb_srcptr a, long p, acb_srcptr b, long q, const acb_t z, ulong n);
|
||||
|
||||
long acb_hypgeom_pfq_choose_n(acb_srcptr a, long p,
|
||||
acb_srcptr b, long q, const acb_t z, long prec);
|
||||
|
||||
void acb_hypgeom_pfq_sum_forward(acb_t s, acb_t t, acb_srcptr a, long p, acb_srcptr b, long q,
|
||||
const acb_t z, long n, long prec);
|
||||
|
||||
void acb_hypgeom_pfq_sum_rs(acb_t s, acb_t t, acb_srcptr a, long p, acb_srcptr b, long q,
|
||||
const acb_t z, long n, long prec);
|
||||
|
||||
void acb_hypgeom_pfq_sum(acb_t s, acb_t t, acb_srcptr a, long p, acb_srcptr b, long q,
|
||||
const acb_t z, long n, long prec);
|
||||
|
||||
|
|
|
|||
134
acb_hypgeom/pfq_choose_n.c
Normal file
134
acb_hypgeom/pfq_choose_n.c
Normal file
|
|
@ -0,0 +1,134 @@
|
|||
/*=============================================================================
|
||||
|
||||
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) 2014 Fredrik Johansson
|
||||
|
||||
******************************************************************************/
|
||||
|
||||
#include "acb_hypgeom.h"
|
||||
|
||||
double mag_get_log2_d_approx(const mag_t x);
|
||||
|
||||
long
|
||||
acb_hypgeom_pfq_choose_n(acb_srcptr a, long p,
|
||||
acb_srcptr b, long q, const acb_t z, long prec)
|
||||
{
|
||||
long k, n, minimum_n, maximum_n;
|
||||
mag_t zmag;
|
||||
|
||||
/* todo: first test for z = 0 and nonpositive integers */
|
||||
|
||||
double t, u;
|
||||
double log2_z;
|
||||
double log2_term;
|
||||
double log2_factor;
|
||||
double log2_term_max;
|
||||
|
||||
double * are;
|
||||
double * aim;
|
||||
double * bre;
|
||||
double * bim;
|
||||
|
||||
mag_init(zmag);
|
||||
|
||||
are = flint_malloc(sizeof(double) * 2 * (p + q));
|
||||
aim = are + p;
|
||||
bre = aim + p;
|
||||
bim = bre + q;
|
||||
|
||||
acb_get_mag(zmag, z);
|
||||
log2_z = mag_get_log2_d_approx(zmag);
|
||||
|
||||
for (k = 0; k < p; k++)
|
||||
{
|
||||
are[k] = arf_get_d(arb_midref(acb_realref(a + k)), ARF_RND_DOWN);
|
||||
aim[k] = arf_get_d(arb_midref(acb_imagref(a + k)), ARF_RND_DOWN);
|
||||
}
|
||||
|
||||
minimum_n = 1;
|
||||
maximum_n = 50 + 2 * prec;
|
||||
|
||||
for (k = 0; k < q; k++)
|
||||
{
|
||||
bre[k] = arf_get_d(arb_midref(acb_realref(b + k)), ARF_RND_DOWN);
|
||||
bim[k] = arf_get_d(arb_midref(acb_imagref(b + k)), ARF_RND_DOWN);
|
||||
|
||||
if (bre[k] <= 0.25)
|
||||
{
|
||||
minimum_n = FLINT_MAX(minimum_n, 2 - bre[k]);
|
||||
}
|
||||
}
|
||||
|
||||
n = 1;
|
||||
log2_term = 0.0;
|
||||
log2_term_max = log2_term;
|
||||
|
||||
while (n <= maximum_n && minimum_n < maximum_n)
|
||||
{
|
||||
if (log2_term < log2_term_max - prec - 4 && n >= minimum_n)
|
||||
break;
|
||||
|
||||
t = 1.0;
|
||||
|
||||
for (k = 0; k < FLINT_MAX(p, q); k++)
|
||||
{
|
||||
if (k < p)
|
||||
{
|
||||
u = (are[k] + n) * (are[k] + n) + (aim[k] * aim[k]);
|
||||
u = FLINT_ABS(u);
|
||||
|
||||
if (u < 1e-8 || u > 1e100 || t > 1e100)
|
||||
goto somethingstrange;
|
||||
|
||||
t *= u;
|
||||
}
|
||||
|
||||
if (k < q)
|
||||
{
|
||||
u = (bre[k] + n) * (bre[k] + n) + (bim[k] * bim[k]);
|
||||
u = FLINT_ABS(u);
|
||||
|
||||
if (u < 1e-8 || u > 1e100 || t > 1e100)
|
||||
goto somethingstrange;
|
||||
|
||||
t /= u;
|
||||
}
|
||||
}
|
||||
|
||||
log2_factor = 0.5 * log(t) * 1.4426950408889634074 + log2_z;
|
||||
|
||||
/* For asymptotic series, require rapid decay */
|
||||
if (p > q && n >= minimum_n && log2_factor > -0.2)
|
||||
break;
|
||||
|
||||
log2_term += log2_factor;
|
||||
log2_term_max = FLINT_MAX(log2_term_max, log2_term);
|
||||
n++;
|
||||
}
|
||||
|
||||
somethingstrange:
|
||||
|
||||
flint_free(are);
|
||||
mag_clear(zmag);
|
||||
|
||||
return n;
|
||||
}
|
||||
|
||||
|
|
@ -37,7 +37,10 @@ acb_hypgeom_pfq_direct(acb_t res, acb_srcptr a, long p, acb_srcptr b, long q,
|
|||
mag_init(err);
|
||||
mag_init(C);
|
||||
|
||||
acb_hypgeom_pfq_sum(s, t, a, p, b, q, z, n, prec);
|
||||
if (n < 0)
|
||||
n = acb_hypgeom_pfq_choose_n(a, p, b, q, z, prec);
|
||||
|
||||
acb_hypgeom_pfq_sum_rs(s, t, a, p, b, q, z, n, prec);
|
||||
|
||||
if (!acb_is_zero(t))
|
||||
{
|
||||
|
|
|
|||
|
|
@ -26,52 +26,17 @@
|
|||
#include "acb_hypgeom.h"
|
||||
|
||||
void
|
||||
acb_hypgeom_pfq_sum(acb_t s, acb_t t,
|
||||
acb_srcptr a, long p, acb_srcptr b, long q, const acb_t z, long n, long prec)
|
||||
acb_hypgeom_pfq_sum(acb_t s, acb_t t, acb_srcptr a, long p,
|
||||
acb_srcptr b, long q, const acb_t z, long n, long prec)
|
||||
{
|
||||
acb_t u, v;
|
||||
long k, i;
|
||||
|
||||
acb_init(u);
|
||||
acb_init(v);
|
||||
|
||||
acb_zero(s);
|
||||
acb_one(t);
|
||||
|
||||
for (k = 0; k < n && !acb_is_zero(t); k++)
|
||||
if (n > 4 && prec >= 128
|
||||
&& _acb_vec_bits(a, p) * p + _acb_vec_bits(b, q) * q + 10 < prec / 2)
|
||||
{
|
||||
acb_add(s, s, t, prec);
|
||||
|
||||
if (p > 0)
|
||||
{
|
||||
acb_add_ui(u, a, k, prec);
|
||||
|
||||
for (i = 1; i < p; i++)
|
||||
{
|
||||
acb_add_ui(v, a + i, k, prec);
|
||||
acb_mul(u, u, v, prec);
|
||||
}
|
||||
|
||||
acb_mul(t, t, u, prec);
|
||||
}
|
||||
|
||||
if (q > 0)
|
||||
{
|
||||
acb_add_ui(u, b, k, prec);
|
||||
|
||||
for (i = 1; i < q; i++)
|
||||
{
|
||||
acb_add_ui(v, b + i, k, prec);
|
||||
acb_mul(u, u, v, prec);
|
||||
}
|
||||
|
||||
acb_div(t, t, u, prec);
|
||||
}
|
||||
|
||||
acb_mul(t, t, z, prec);
|
||||
acb_hypgeom_pfq_sum_rs(s, t, a, p, b, q, z, n, prec);
|
||||
}
|
||||
else
|
||||
{
|
||||
acb_hypgeom_pfq_sum_forward(s, t, a, p, b, q, z, n, prec);
|
||||
}
|
||||
|
||||
acb_clear(u);
|
||||
acb_clear(v);
|
||||
}
|
||||
|
||||
|
|
|
|||
77
acb_hypgeom/pfq_sum_forward.c
Normal file
77
acb_hypgeom/pfq_sum_forward.c
Normal file
|
|
@ -0,0 +1,77 @@
|
|||
/*=============================================================================
|
||||
|
||||
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) 2014 Fredrik Johansson
|
||||
|
||||
******************************************************************************/
|
||||
|
||||
#include "acb_hypgeom.h"
|
||||
|
||||
void
|
||||
acb_hypgeom_pfq_sum_forward(acb_t s, acb_t t,
|
||||
acb_srcptr a, long p, acb_srcptr b, long q, const acb_t z, long n, long prec)
|
||||
{
|
||||
acb_t u, v;
|
||||
long k, i;
|
||||
|
||||
acb_init(u);
|
||||
acb_init(v);
|
||||
|
||||
acb_zero(s);
|
||||
acb_one(t);
|
||||
|
||||
for (k = 0; k < n && !acb_is_zero(t); k++)
|
||||
{
|
||||
acb_add(s, s, t, prec);
|
||||
|
||||
if (p > 0)
|
||||
{
|
||||
acb_add_ui(u, a, k, prec);
|
||||
|
||||
for (i = 1; i < p; i++)
|
||||
{
|
||||
acb_add_ui(v, a + i, k, prec);
|
||||
acb_mul(u, u, v, prec);
|
||||
}
|
||||
|
||||
acb_mul(t, t, u, prec);
|
||||
}
|
||||
|
||||
if (q > 0)
|
||||
{
|
||||
acb_add_ui(u, b, k, prec);
|
||||
|
||||
for (i = 1; i < q; i++)
|
||||
{
|
||||
acb_add_ui(v, b + i, k, prec);
|
||||
acb_mul(u, u, v, prec);
|
||||
}
|
||||
|
||||
acb_div(t, t, u, prec);
|
||||
}
|
||||
|
||||
acb_mul(t, t, z, prec);
|
||||
}
|
||||
|
||||
acb_clear(u);
|
||||
acb_clear(v);
|
||||
}
|
||||
|
||||
130
acb_hypgeom/pfq_sum_rs.c
Normal file
130
acb_hypgeom/pfq_sum_rs.c
Normal file
|
|
@ -0,0 +1,130 @@
|
|||
/*=============================================================================
|
||||
|
||||
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) 2014 Fredrik Johansson
|
||||
|
||||
******************************************************************************/
|
||||
|
||||
#include "acb_hypgeom.h"
|
||||
|
||||
void
|
||||
acb_hypgeom_pfq_sum_rs(acb_t res, acb_t term, acb_srcptr a, long p,
|
||||
acb_srcptr b, long q, const acb_t z, long n, long prec)
|
||||
{
|
||||
acb_ptr zpow;
|
||||
acb_t s, t, u;
|
||||
long i, j, k, m;
|
||||
mag_t B, C;
|
||||
|
||||
if (n == 0)
|
||||
{
|
||||
acb_zero(res);
|
||||
acb_one(term);
|
||||
return;
|
||||
}
|
||||
|
||||
if (n < 0)
|
||||
abort();
|
||||
|
||||
m = n_sqrt(n);
|
||||
m = FLINT_MIN(m, 150);
|
||||
|
||||
mag_init(B);
|
||||
mag_init(C);
|
||||
acb_init(s);
|
||||
acb_init(t);
|
||||
acb_init(u);
|
||||
zpow = _acb_vec_init(m + 1);
|
||||
|
||||
_acb_vec_set_powers(zpow, z, m + 1, prec);
|
||||
|
||||
mag_one(B);
|
||||
|
||||
for (k = n; k >= 0; k--)
|
||||
{
|
||||
j = k % m;
|
||||
|
||||
if (k < n)
|
||||
acb_add(s, s, zpow + j, prec);
|
||||
|
||||
if (k > 0)
|
||||
{
|
||||
if (p > 0)
|
||||
{
|
||||
acb_add_ui(u, a, k - 1, prec);
|
||||
|
||||
for (i = 1; i < p; i++)
|
||||
{
|
||||
acb_add_ui(t, a + i, k - 1, prec);
|
||||
acb_mul(u, u, t, prec);
|
||||
}
|
||||
|
||||
if (k < n)
|
||||
acb_mul(s, s, u, prec);
|
||||
|
||||
acb_get_mag(C, u);
|
||||
mag_mul(B, B, C);
|
||||
}
|
||||
|
||||
if (q > 0)
|
||||
{
|
||||
acb_add_ui(u, b, k - 1, prec);
|
||||
|
||||
for (i = 1; i < q; i++)
|
||||
{
|
||||
acb_add_ui(t, b + i, k - 1, prec);
|
||||
acb_mul(u, u, t, prec);
|
||||
}
|
||||
|
||||
if (k < n)
|
||||
acb_div(s, s, u, prec);
|
||||
|
||||
acb_get_mag_lower(C, u);
|
||||
mag_div(B, B, C);
|
||||
}
|
||||
|
||||
if (j == 0 && k < n)
|
||||
{
|
||||
acb_mul(s, s, zpow + m, prec);
|
||||
}
|
||||
}
|
||||
}
|
||||
|
||||
acb_get_mag(C, z);
|
||||
mag_pow_ui(C, C, n);
|
||||
mag_mul(B, B, C);
|
||||
|
||||
acb_zero(term);
|
||||
if (_acb_vec_is_real(a, p) && _acb_vec_is_real(b, q) && acb_is_real(z))
|
||||
arb_add_error_mag(acb_realref(term), B);
|
||||
else
|
||||
acb_add_error_mag(term, B);
|
||||
|
||||
acb_set(res, s);
|
||||
|
||||
mag_clear(B);
|
||||
mag_clear(C);
|
||||
acb_clear(s);
|
||||
acb_clear(t);
|
||||
acb_clear(u);
|
||||
_acb_vec_clear(zpow, m + 1);
|
||||
}
|
||||
|
||||
|
|
@ -178,6 +178,8 @@ void acb_hypgeom_u_asymp(acb_t res, const acb_t a, const acb_t b,
|
|||
const acb_t z, long n, long prec)
|
||||
{
|
||||
mag_t C1, Cn, alpha, nu, sigma, rho, zinv, tmp, err;
|
||||
acb_struct aa[3];
|
||||
acb_t s, t, w;
|
||||
int R;
|
||||
|
||||
if (!acb_is_finite(a) || !acb_is_finite(b) || !acb_is_finite(z))
|
||||
|
|
@ -196,6 +198,24 @@ void acb_hypgeom_u_asymp(acb_t res, const acb_t a, const acb_t b,
|
|||
mag_init(tmp);
|
||||
mag_init(err);
|
||||
|
||||
acb_init(aa);
|
||||
acb_init(aa + 1);
|
||||
acb_init(aa + 2);
|
||||
acb_init(s);
|
||||
acb_init(t);
|
||||
acb_init(w);
|
||||
|
||||
acb_set(aa, a);
|
||||
acb_sub(aa + 1, a, b, prec);
|
||||
acb_add_ui(aa + 1, aa + 1, 1, prec);
|
||||
acb_one(aa + 2);
|
||||
|
||||
acb_neg(w, z);
|
||||
acb_inv(w, w, prec);
|
||||
|
||||
if (n < 0)
|
||||
n = acb_hypgeom_pfq_choose_n(aa, 2, aa + 2, 1, w, prec);
|
||||
|
||||
acb_hypgeom_u_asymp_bound_factors(&R, alpha, nu,
|
||||
sigma, rho, zinv, a, b, z);
|
||||
|
||||
|
|
@ -205,6 +225,8 @@ void acb_hypgeom_u_asymp(acb_t res, const acb_t a, const acb_t b,
|
|||
}
|
||||
else
|
||||
{
|
||||
acb_hypgeom_pfq_sum(s, t, aa, 2, aa + 2, 1, w, n, prec);
|
||||
|
||||
if (R == 1)
|
||||
{
|
||||
mag_one(C1);
|
||||
|
|
@ -239,54 +261,12 @@ void acb_hypgeom_u_asymp(acb_t res, const acb_t a, const acb_t b,
|
|||
mag_mul(err, err, alpha);
|
||||
mag_mul_2exp_si(err, err, 1);
|
||||
|
||||
/* evaluate the sum, naively for now */
|
||||
{
|
||||
acb_t s, t, u, v, w, ab1;
|
||||
long k;
|
||||
/* nth term * factor */
|
||||
acb_get_mag(tmp, t);
|
||||
mag_mul(err, err, tmp);
|
||||
acb_add_error_mag(s, err);
|
||||
|
||||
acb_init(s);
|
||||
acb_init(t);
|
||||
acb_init(u);
|
||||
acb_init(v);
|
||||
acb_init(w);
|
||||
acb_init(ab1);
|
||||
|
||||
acb_one(t);
|
||||
acb_sub(ab1, a, b, prec);
|
||||
acb_add_ui(ab1, ab1, 1, prec);
|
||||
|
||||
if (n != 0)
|
||||
{
|
||||
acb_neg(w, z);
|
||||
acb_inv(w, w, prec);
|
||||
}
|
||||
|
||||
for (k = 0; k < n && !acb_is_zero(t); k++)
|
||||
{
|
||||
acb_add(s, s, t, prec);
|
||||
|
||||
acb_add_ui(u, a, k, prec);
|
||||
acb_add_ui(v, ab1, k, prec);
|
||||
acb_mul(u, u, v, prec);
|
||||
acb_mul(t, t, u, prec);
|
||||
acb_mul(t, t, w, prec);
|
||||
acb_div_ui(t, t, k + 1, prec);
|
||||
}
|
||||
|
||||
/* nth term */
|
||||
acb_get_mag(tmp, t);
|
||||
mag_mul(err, err, tmp);
|
||||
acb_add_error_mag(s, err);
|
||||
|
||||
acb_set(res, s);
|
||||
|
||||
acb_clear(s);
|
||||
acb_clear(t);
|
||||
acb_clear(u);
|
||||
acb_clear(v);
|
||||
acb_clear(w);
|
||||
acb_clear(ab1);
|
||||
}
|
||||
acb_set(res, s);
|
||||
}
|
||||
|
||||
mag_clear(C1);
|
||||
|
|
@ -298,5 +278,12 @@ void acb_hypgeom_u_asymp(acb_t res, const acb_t a, const acb_t b,
|
|||
mag_clear(zinv);
|
||||
mag_clear(tmp);
|
||||
mag_clear(err);
|
||||
|
||||
acb_clear(aa);
|
||||
acb_clear(aa + 1);
|
||||
acb_clear(aa + 2);
|
||||
acb_clear(s);
|
||||
acb_clear(t);
|
||||
acb_clear(w);
|
||||
}
|
||||
|
||||
|
|
|
|||
|
|
@ -66,13 +66,42 @@ or remove a 1 from the `a_i` parameters if there is one.
|
|||
As currently implemented, the bound becomes infinite when `n` is
|
||||
too small, even if the series converges.
|
||||
|
||||
.. function:: long acb_hypgeom_pfq_choose_n(acb_srcptr a, long p, acb_srcptr b, long q, const acb_t z, long prec)
|
||||
|
||||
Heuristically attempts to choose a number of terms *n* to
|
||||
sum of a hypergeometric series at a working precision of *prec* bits.
|
||||
|
||||
Uses double precision arithmetic internally. As currently implemented,
|
||||
it can fail to produce a good result if the parameters are extremely
|
||||
large or extremely close to nonpositive integers.
|
||||
|
||||
Numerical cancellation is assumed to be significant, so truncation
|
||||
is done when the current term is *prec* bits
|
||||
smaller than the largest encountered term.
|
||||
|
||||
This function will also attempt to pick a reasonable
|
||||
truncation point for divergent series.
|
||||
|
||||
.. function:: void acb_hypgeom_pfq_sum_forward(acb_t s, acb_t t, acb_srcptr a, long p, acb_srcptr b, long q, const acb_t z, long n, long prec)
|
||||
|
||||
.. function:: void acb_hypgeom_pfq_sum_rs(acb_t s, acb_t t, acb_srcptr a, long p, acb_srcptr b, long q, const acb_t z, long n, long prec)
|
||||
|
||||
.. function:: void acb_hypgeom_pfq_sum(acb_t s, acb_t t, acb_srcptr a, long p, acb_srcptr b, long q, const acb_t z, long n, long prec)
|
||||
|
||||
Computes `s = \sum_{k=0}^{n-1} T(k)` and `t = T(n)`.
|
||||
|
||||
Does not allow aliasing between input and output variables.
|
||||
We require `n \ge 0`.
|
||||
|
||||
The *forward* version computes the sum using forward
|
||||
recurrence.
|
||||
|
||||
The *rs* version computes the sum in reverse order
|
||||
using rectangular splitting. It only computes a
|
||||
magnitude bound for the value of *t*.
|
||||
|
||||
The default version automatically chooses an algorithm
|
||||
depending on the inputs.
|
||||
|
||||
.. function:: void acb_hypgeom_pfq_direct(acb_t res, acb_srcptr a, long p, acb_srcptr b, long q, const acb_t z, long n, long prec)
|
||||
|
||||
Computes
|
||||
|
|
@ -85,7 +114,9 @@ or remove a 1 from the `a_i` parameters if there is one.
|
|||
|
||||
directly from the defining series, including a rigorous bound for
|
||||
the truncation error `\varepsilon` in the output.
|
||||
We require `n \ge 0`.
|
||||
|
||||
If `n < 0`, this function chooses a number of terms automatically
|
||||
using :func:`acb_hypgeom_pfq_choose_n`.
|
||||
|
||||
Asymptotic series
|
||||
-------------------------------------------------------------------------------
|
||||
|
|
|
|||
Loading…
Add table
Reference in a new issue