/* specfunc/coupling.c
*
* Copyright (C) 1996, 1997, 1998, 1999, 2000, 2001, 2002 Gerard Jungman
*
* This program 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 3 of the License, or (at
* your option) any later version.
*
* This program 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 this program; if not, write to the Free Software
* Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301, USA.
*/
/* Author: G. Jungman */
#include <config.h>
#include <stdlib.h>
#include <gsl/gsl_math.h>
#include <gsl/gsl_errno.h>
#include <gsl/gsl_sf_gamma.h>
#include <gsl/gsl_sf_coupling.h>
#include <gsl/gsl_sf_exp.h>
#include "error.h"
inline
static
int locMax3(const int a, const int b, const int c)
{
int d = GSL_MAX(a, b);
return GSL_MAX(d, c);
}
inline
static
int locMin3(const int a, const int b, const int c)
{
int d = GSL_MIN(a, b);
return GSL_MIN(d, c);
}
inline
static
int locMin5(const int a, const int b, const int c, const int d, const int e)
{
int f = GSL_MIN(a, b);
int g = GSL_MIN(c, d);
int h = GSL_MIN(f, g);
return GSL_MIN(e, h);
}
/* See: [Thompson, Atlas for Computing Mathematical Functions] */
static
int
delta(int ta, int tb, int tc, gsl_sf_result * d)
{
gsl_sf_result f1, f2, f3, f4;
int status = 0;
status += gsl_sf_fact_e((ta + tb - tc)/2, &f1);
status += gsl_sf_fact_e((ta + tc - tb)/2, &f2);
status += gsl_sf_fact_e((tb + tc - ta)/2, &f3);
status += gsl_sf_fact_e((ta + tb + tc)/2 + 1, &f4);
if(status != 0) {
OVERFLOW_ERROR(d);
}
d->val = f1.val * f2.val * f3.val / f4.val;
d->err = 4.0 * GSL_DBL_EPSILON * fabs(d->val);
return GSL_SUCCESS;
}
static
int
triangle_selection_fails(int two_ja, int two_jb, int two_jc)
{
/*
* enough to check the triangle condition for one spin vs. the other two
*/
return ( (two_jb < abs(two_ja - two_jc)) || (two_jb > two_ja + two_jc) ||
GSL_IS_ODD(two_ja + two_jb + two_jc) );
}
static
int
m_selection_fails(int two_ja, int two_jb, int two_jc,
int two_ma, int two_mb, int two_mc)
{
return (
abs(two_ma) > two_ja
|| abs(two_mb) > two_jb
|| abs(two_mc) > two_jc
|| GSL_IS_ODD(two_ja + two_ma)
|| GSL_IS_ODD(two_jb + two_mb)
|| GSL_IS_ODD(two_jc + two_mc)
|| (two_ma + two_mb + two_mc) != 0
);
}
/*-*-*-*-*-*-*-*-*-*-*-* Functions with Error Codes *-*-*-*-*-*-*-*-*-*-*-*/
int
gsl_sf_coupling_3j_e (int two_ja, int two_jb, int two_jc,
int two_ma, int two_mb, int two_mc,
gsl_sf_result * result)
{
/* CHECK_POINTER(result) */
if(two_ja < 0 || two_jb < 0 || two_jc < 0) {
DOMAIN_ERROR(result);
}
else if ( triangle_selection_fails(two_ja, two_jb, two_jc)
|| m_selection_fails(two_ja, two_jb, two_jc, two_ma, two_mb, two_mc)
) {
result->val = 0.0;
result->err = 0.0;
return GSL_SUCCESS;
}
else if ( two_ma == 0 && two_mb == 0 && two_mc == 0
&& ((two_ja + two_jb + two_jc) % 4 == 2) ) {
/* Special case for (ja jb jc; 0 0 0) = 0 when ja+jb+jc=odd */
result->val = 0.0;
result->err = 0.0;
return GSL_SUCCESS;
}
else {
int jca = (-two_ja + two_jb + two_jc) / 2,
jcb = ( two_ja - two_jb + two_jc) / 2,
jcc = ( two_ja + two_jb - two_jc) / 2,
jmma = ( two_ja - two_ma) / 2,
jmmb = ( two_jb - two_mb) / 2,
jmmc = ( two_jc - two_mc) / 2,
jpma = ( two_ja + two_ma) / 2,
jpmb = ( two_jb + two_mb) / 2,
jpmc = ( two_jc + two_mc) / 2,
jsum = ( two_ja + two_jb + two_jc) / 2,
kmin = locMax3 (0, jpmb - jmmc, jmma - jpmc),
kmax = locMin3 (jcc, jmma, jpmb),
k, sign = GSL_IS_ODD (kmin - jpma + jmmb) ? -1 : 1,
status = 0;
double sum_pos = 0.0, sum_neg = 0.0, sum_err = 0.0;
gsl_sf_result bc1, bc2, bc3, bcn1, bcn2, bcd1, bcd2, bcd3, bcd4, term, lnorm;
status += gsl_sf_lnchoose_e (two_ja, jcc , &bcn1);
status += gsl_sf_lnchoose_e (two_jb, jcc , &bcn2);
status += gsl_sf_lnchoose_e (jsum+1, jcc , &bcd1);
status += gsl_sf_lnchoose_e (two_ja, jmma, &bcd2);
status += gsl_sf_lnchoose_e (two_jb, jmmb, &bcd3);
status += gsl_sf_lnchoose_e (two_jc, jpmc, &bcd4);
lnorm.val = 0.5 * (bcn1.val + bcn2.val - bcd1.val - bcd2.val - bcd3.val
- bcd4.val - log(two_jc + 1.0));
lnorm.err = 0.5 * (bcn1.err + bcn2.err + bcd1.err + bcd2.err + bcd3.err
+ bcd4.err + GSL_DBL_EPSILON * log(two_jc + 1.0));
for (k = kmin; k <= kmax; k++) {
status += gsl_sf_lnchoose_e (jcc, k, &bc1);
status += gsl_sf_lnchoose_e (jcb, jmma - k, &bc2);
status += gsl_sf_lnchoose_e (jca, jpmb - k, &bc3);
status += gsl_sf_exp_err_e(bc1.val + bc2.val + bc3.val + lnorm.val,
bc1.err + bc2.err + bc3.err + lnorm.err,
&term);
if (status != 0) {
OVERFLOW_ERROR (result);
}
if (sign < 0) {
sum_neg += term.val;
} else {
sum_pos += term.val;
}
sum_err += term.err;
sign = -sign;
}
result->val = sum_pos - sum_neg;
result->err = sum_err;
result->err += 2.0 * GSL_DBL_EPSILON * (sum_pos + sum_neg);
result->err += 2.0 * GSL_DBL_EPSILON * (kmax - kmin) * fabs(result->val);
return GSL_SUCCESS;
}
}
#ifndef GSL_DISABLE_DEPRECATED
int
gsl_sf_coupling_6j_INCORRECT_e(int two_ja, int two_jb, int two_jc,
int two_jd, int two_je, int two_jf,
gsl_sf_result * result)
{
return gsl_sf_coupling_6j_e(two_ja, two_jb, two_je, two_jd, two_jc, two_jf, result);
}
#endif
int
gsl_sf_coupling_6j_e(int two_ja, int two_jb, int two_jc,
int two_jd, int two_je, int two_jf,
gsl_sf_result * result)
{
/* CHECK_POINTER(result) */
if( two_ja < 0 || two_jb < 0 || two_jc < 0
|| two_jd < 0 || two_je < 0 || two_jf < 0
) {
DOMAIN_ERROR(result);
}
else if( triangle_selection_fails(two_ja, two_jb, two_jc)
|| triangle_selection_fails(two_ja, two_je, two_jf)
|| triangle_selection_fails(two_jb, two_jd, two_jf)
|| triangle_selection_fails(two_je, two_jd, two_jc)
) {
result->val = 0.0;
result->err = 0.0;
return GSL_SUCCESS;
}
else {
gsl_sf_result n1;
gsl_sf_result d1, d2, d3, d4, d5, d6;
double norm;
int tk, tkmin, tkmax;
double phase;
double sum_pos = 0.0;
double sum_neg = 0.0;
double sumsq_err = 0.0;
int status = 0;
status += delta(two_ja, two_jb, two_jc, &d1);
status += delta(two_ja, two_je, two_jf, &d2);
status += delta(two_jb, two_jd, two_jf, &d3);
status += delta(two_je, two_jd, two_jc, &d4);
if(status != GSL_SUCCESS) {
OVERFLOW_ERROR(result);
}
norm = sqrt(d1.val) * sqrt(d2.val) * sqrt(d3.val) * sqrt(d4.val);
tkmin = locMax3(0,
two_ja + two_jd - two_jc - two_jf,
two_jb + two_je - two_jc - two_jf);
tkmax = locMin5(two_ja + two_jb + two_je + two_jd + 2,
two_ja + two_jb - two_jc,
two_je + two_jd - two_jc,
two_ja + two_je - two_jf,
two_jb + two_jd - two_jf);
phase = GSL_IS_ODD((two_ja + two_jb + two_je + two_jd + tkmin)/2)
? -1.0
: 1.0;
for(tk=tkmin; tk<=tkmax; tk += 2) {
double term;
double term_err;
gsl_sf_result den_1, den_2;
gsl_sf_result d1_a, d1_b;
status = 0;
status += gsl_sf_fact_e((two_ja + two_jb + two_je + two_jd - tk)/2 + 1, &n1);
status += gsl_sf_fact_e(tk/2, &d1_a);
status += gsl_sf_fact_e((two_jc + two_jf - two_ja - two_jd + tk)/2, &d1_b);
status += gsl_sf_fact_e((two_jc + two_jf - two_jb - two_je + tk)/2, &d2);
status += gsl_sf_fact_e((two_ja + two_jb - two_jc - tk)/2, &d3);
status += gsl_sf_fact_e((two_je + two_jd - two_jc - tk)/2, &d4);
status += gsl_sf_fact_e((two_ja + two_je - two_jf - tk)/2, &d5);
status += gsl_sf_fact_e((two_jb + two_jd - two_jf - tk)/2, &d6);
if(status != GSL_SUCCESS) {
OVERFLOW_ERROR(result);
}
d1.val = d1_a.val * d1_b.val;
d1.err = d1_a.err * fabs(d1_b.val) + fabs(d1_a.val) * d1_b.err;
den_1.val = d1.val*d2.val*d3.val;
den_1.err = d1.err * fabs(d2.val*d3.val);
den_1.err += d2.err * fabs(d1.val*d3.val);
den_1.err += d3.err * fabs(d1.val*d2.val);
den_2.val = d4.val*d5.val*d6.val;
den_2.err = d4.err * fabs(d5.val*d6.val);
den_2.err += d5.err * fabs(d4.val*d6.val);
den_2.err += d6.err * fabs(d4.val*d5.val);
term = phase * n1.val / den_1.val / den_2.val;
phase = -phase;
term_err = n1.err / fabs(den_1.val) / fabs(den_2.val);
term_err += fabs(term / den_1.val) * den_1.err;
term_err += fabs(term / den_2.val) * den_2.err;
if(term >= 0.0) {
sum_pos += norm*term;
}
else {
sum_neg -= norm*term;
}
sumsq_err += norm*norm * term_err*term_err;
}
result->val = sum_pos - sum_neg;
result->err = 2.0 * GSL_DBL_EPSILON * (sum_pos + sum_neg);
result->err += sqrt(sumsq_err / (0.5*(tkmax-tkmin)+1.0));
result->err += 2.0 * GSL_DBL_EPSILON * (tkmax - tkmin + 2.0) * fabs(result->val);
return GSL_SUCCESS;
}
}
int
gsl_sf_coupling_RacahW_e(int two_ja, int two_jb, int two_jc,
int two_jd, int two_je, int two_jf,
gsl_sf_result * result)
{
int status = gsl_sf_coupling_6j_e(two_ja, two_jb, two_je, two_jd, two_jc, two_jf, result);
int phase_sum = (two_ja + two_jb + two_jc + two_jd)/2;
result->val *= ( GSL_IS_ODD(phase_sum) ? -1.0 : 1.0 );
return status;
}
int
gsl_sf_coupling_9j_e(int two_ja, int two_jb, int two_jc,
int two_jd, int two_je, int two_jf,
int two_jg, int two_jh, int two_ji,
gsl_sf_result * result)
{
/* CHECK_POINTER(result) */
if( two_ja < 0 || two_jb < 0 || two_jc < 0
|| two_jd < 0 || two_je < 0 || two_jf < 0
|| two_jg < 0 || two_jh < 0 || two_ji < 0
) {
DOMAIN_ERROR(result);
}
else if( triangle_selection_fails(two_ja, two_jb, two_jc)
|| triangle_selection_fails(two_jd, two_je, two_jf)
|| triangle_selection_fails(two_jg, two_jh, two_ji)
|| triangle_selection_fails(two_ja, two_jd, two_jg)
|| triangle_selection_fails(two_jb, two_je, two_jh)
|| triangle_selection_fails(two_jc, two_jf, two_ji)
) {
result->val = 0.0;
result->err = 0.0;
return GSL_SUCCESS;
}
else {
int tk;
int tkmin = locMax3(abs(two_ja-two_ji), abs(two_jh-two_jd), abs(two_jb-two_jf));
int tkmax = locMin3(two_ja + two_ji, two_jh + two_jd, two_jb + two_jf);
double sum_pos = 0.0;
double sum_neg = 0.0;
double sumsq_err = 0.0;
double phase;
for(tk=tkmin; tk<=tkmax; tk += 2) {
gsl_sf_result s1, s2, s3;
double term;
double term_err;
int status = 0;
status += gsl_sf_coupling_6j_e(two_ja, two_ji, tk, two_jh, two_jd, two_jg, &s1);
status += gsl_sf_coupling_6j_e(two_jb, two_jf, tk, two_jd, two_jh, two_je, &s2);
status += gsl_sf_coupling_6j_e(two_ja, two_ji, tk, two_jf, two_jb, two_jc, &s3);
if(status != GSL_SUCCESS) {
OVERFLOW_ERROR(result);
}
term = s1.val * s2.val * s3.val;
term_err = s1.err * fabs(s2.val*s3.val);
term_err += s2.err * fabs(s1.val*s3.val);
term_err += s3.err * fabs(s1.val*s2.val);
if(term >= 0.0) {
sum_pos += (tk + 1) * term;
}
else {
sum_neg -= (tk + 1) * term;
}
sumsq_err += ((tk+1) * term_err) * ((tk+1) * term_err);
}
phase = GSL_IS_ODD(tkmin) ? -1.0 : 1.0;
result->val = phase * (sum_pos - sum_neg);
result->err = 2.0 * GSL_DBL_EPSILON * (sum_pos + sum_neg);
result->err += sqrt(sumsq_err / (0.5*(tkmax-tkmin)+1.0));
result->err += 2.0 * GSL_DBL_EPSILON * (tkmax-tkmin + 2.0) * fabs(result->val);
return GSL_SUCCESS;
}
}
/*-*-*-*-*-*-*-*-*-* Functions w/ Natural Prototypes *-*-*-*-*-*-*-*-*-*-*/
#include "eval.h"
double gsl_sf_coupling_3j(int two_ja, int two_jb, int two_jc,
int two_ma, int two_mb, int two_mc)
{
EVAL_RESULT(gsl_sf_coupling_3j_e(two_ja, two_jb, two_jc,
two_ma, two_mb, two_mc,
&result));
}
#ifndef GSL_DISABLE_DEPRECATED
double gsl_sf_coupling_6j_INCORRECT(int two_ja, int two_jb, int two_jc,
int two_jd, int two_je, int two_jf)
{
EVAL_RESULT(gsl_sf_coupling_6j_INCORRECT_e(two_ja, two_jb, two_jc,
two_jd, two_je, two_jf,
&result));
}
#endif
double gsl_sf_coupling_6j(int two_ja, int two_jb, int two_jc,
int two_jd, int two_je, int two_jf)
{
EVAL_RESULT(gsl_sf_coupling_6j_e(two_ja, two_jb, two_jc,
two_jd, two_je, two_jf,
&result));
}
double gsl_sf_coupling_RacahW(int two_ja, int two_jb, int two_jc,
int two_jd, int two_je, int two_jf)
{
EVAL_RESULT(gsl_sf_coupling_RacahW_e(two_ja, two_jb, two_jc,
two_jd, two_je, two_jf,
&result));
}
double gsl_sf_coupling_9j(int two_ja, int two_jb, int two_jc,
int two_jd, int two_je, int two_jf,
int two_jg, int two_jh, int two_ji)
{
EVAL_RESULT(gsl_sf_coupling_9j_e(two_ja, two_jb, two_jc,
two_jd, two_je, two_jf,
two_jg, two_jh, two_ji,
&result));
}