/* specfunc/legendre_Qn.c
*
* Copyright (C) 1996, 1997, 1998, 1999, 2000 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 <gsl/gsl_math.h>
#include <gsl/gsl_errno.h>
#include <gsl/gsl_sf_bessel.h>
#include <gsl/gsl_sf_elementary.h>
#include <gsl/gsl_sf_exp.h>
#include <gsl/gsl_sf_pow_int.h>
#include <gsl/gsl_sf_legendre.h>
#include "error.h"
/* Evaluate f_{ell+1}/f_ell
* f_ell := Q^{b}_{a+ell}(x)
* x > 1
*/
static
int
legendreQ_CF1_xgt1(int ell, double a, double b, double x, double * result)
{
const double RECUR_BIG = GSL_SQRT_DBL_MAX;
const int maxiter = 5000;
int n = 1;
double Anm2 = 1.0;
double Bnm2 = 0.0;
double Anm1 = 0.0;
double Bnm1 = 1.0;
double a1 = ell + 1.0 + a + b;
double b1 = (2.0*(ell+1.0+a) + 1.0) * x;
double An = b1*Anm1 + a1*Anm2;
double Bn = b1*Bnm1 + a1*Bnm2;
double an, bn;
double fn = An/Bn;
while(n < maxiter) {
double old_fn;
double del;
double lna;
n++;
Anm2 = Anm1;
Bnm2 = Bnm1;
Anm1 = An;
Bnm1 = Bn;
lna = ell + n + a;
an = b*b - lna*lna;
bn = (2.0*lna + 1.0) * x;
An = bn*Anm1 + an*Anm2;
Bn = bn*Bnm1 + an*Bnm2;
if(fabs(An) > RECUR_BIG || fabs(Bn) > RECUR_BIG) {
An /= RECUR_BIG;
Bn /= RECUR_BIG;
Anm1 /= RECUR_BIG;
Bnm1 /= RECUR_BIG;
Anm2 /= RECUR_BIG;
Bnm2 /= RECUR_BIG;
}
old_fn = fn;
fn = An/Bn;
del = old_fn/fn;
if(fabs(del - 1.0) < 4.0*GSL_DBL_EPSILON) break;
}
*result = fn;
if(n == maxiter)
GSL_ERROR ("error", GSL_EMAXITER);
else
return GSL_SUCCESS;
}
/* Uniform asymptotic for Q_l(x).
* Assumes x > -1.0 and x != 1.0.
* Discards second order and higher terms.
*/
static
int
legendre_Ql_asymp_unif(const double ell, const double x, gsl_sf_result * result)
{
if(x < 1.0) {
double u = ell + 0.5;
double th = acos(x);
gsl_sf_result Y0, Y1;
int stat_Y0, stat_Y1;
int stat_m;
double pre;
double B00;
double sum;
/* B00 = 1/8 (1 - th cot(th) / th^2
* pre = sqrt(th/sin(th))
*/
if(th < GSL_ROOT4_DBL_EPSILON) {
B00 = (1.0 + th*th/15.0)/24.0;
pre = 1.0 + th*th/12.0;
}
else {
double sin_th = sqrt(1.0 - x*x);
double cot_th = x / sin_th;
B00 = 1.0/8.0 * (1.0 - th * cot_th) / (th*th);
pre = sqrt(th/sin_th);
}
stat_Y0 = gsl_sf_bessel_Y0_e(u*th, &Y0);
stat_Y1 = gsl_sf_bessel_Y1_e(u*th, &Y1);
sum = -0.5*M_PI * (Y0.val + th/u * Y1.val * B00);
stat_m = gsl_sf_multiply_e(pre, sum, result);
result->err += 0.5*M_PI * fabs(pre) * (Y0.err + fabs(th/u*B00)*Y1.err);
result->err += GSL_DBL_EPSILON * fabs(result->val);
return GSL_ERROR_SELECT_3(stat_m, stat_Y0, stat_Y1);
}
else {
double u = ell + 0.5;
double xi = acosh(x);
gsl_sf_result K0_scaled, K1_scaled;
int stat_K0, stat_K1;
int stat_e;
double pre;
double B00;
double sum;
/* B00 = -1/8 (1 - xi coth(xi) / xi^2
* pre = sqrt(xi/sinh(xi))
*/
if(xi < GSL_ROOT4_DBL_EPSILON) {
B00 = (1.0-xi*xi/15.0)/24.0;
pre = 1.0 - xi*xi/12.0;
}
else {
double sinh_xi = sqrt(x*x - 1.0);
double coth_xi = x / sinh_xi;
B00 = -1.0/8.0 * (1.0 - xi * coth_xi) / (xi*xi);
pre = sqrt(xi/sinh_xi);
}
stat_K0 = gsl_sf_bessel_K0_scaled_e(u*xi, &K0_scaled);
stat_K1 = gsl_sf_bessel_K1_scaled_e(u*xi, &K1_scaled);
sum = K0_scaled.val - xi/u * K1_scaled.val * B00;
stat_e = gsl_sf_exp_mult_e(-u*xi, pre * sum, result);
result->err = GSL_DBL_EPSILON * fabs(result->val) * fabs(u*xi);
result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
return GSL_ERROR_SELECT_3(stat_e, stat_K0, stat_K1);
}
}
/*-*-*-*-*-*-*-*-*-*-*-* Functions with Error Codes *-*-*-*-*-*-*-*-*-*-*-*/
int
gsl_sf_legendre_Q0_e(const double x, gsl_sf_result * result)
{
/* CHECK_POINTER(result) */
if(x <= -1.0 || x == 1.0) {
DOMAIN_ERROR(result);
}
else if(x*x < GSL_ROOT6_DBL_EPSILON) { /* |x| <~ 0.05 */
const double c3 = 1.0/3.0;
const double c5 = 1.0/5.0;
const double c7 = 1.0/7.0;
const double c9 = 1.0/9.0;
const double c11 = 1.0/11.0;
const double y = x * x;
const double series = 1.0 + y*(c3 + y*(c5 + y*(c7 + y*(c9 + y*c11))));
result->val = x * series;
result->err = 2.0 * GSL_DBL_EPSILON * fabs(x);
return GSL_SUCCESS;
}
else if(x < 1.0) {
result->val = 0.5 * log((1.0+x)/(1.0-x));
result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val);
return GSL_SUCCESS;
}
else if(x < 10.0) {
result->val = 0.5 * log((x+1.0)/(x-1.0));
result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val);
return GSL_SUCCESS;
}
else if(x*GSL_DBL_MIN < 2.0) {
const double y = 1.0/(x*x);
const double c1 = 1.0/3.0;
const double c2 = 1.0/5.0;
const double c3 = 1.0/7.0;
const double c4 = 1.0/9.0;
const double c5 = 1.0/11.0;
const double c6 = 1.0/13.0;
const double c7 = 1.0/15.0;
result->val = (1.0/x) * (1.0 + y*(c1 + y*(c2 + y*(c3 + y*(c4 + y*(c5 + y*(c6 + y*c7)))))));
result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val);
return GSL_SUCCESS;
}
else {
UNDERFLOW_ERROR(result);
}
}
int
gsl_sf_legendre_Q1_e(const double x, gsl_sf_result * result)
{
/* CHECK_POINTER(result) */
if(x <= -1.0 || x == 1.0) {
DOMAIN_ERROR(result);
}
else if(x*x < GSL_ROOT6_DBL_EPSILON) { /* |x| <~ 0.05 */
const double c3 = 1.0/3.0;
const double c5 = 1.0/5.0;
const double c7 = 1.0/7.0;
const double c9 = 1.0/9.0;
const double c11 = 1.0/11.0;
const double y = x * x;
const double series = 1.0 + y*(c3 + y*(c5 + y*(c7 + y*(c9 + y*c11))));
result->val = x * x * series - 1.0;
result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val);
return GSL_SUCCESS;
}
else if(x < 1.0){
result->val = 0.5 * x * (log((1.0+x)/(1.0-x))) - 1.0;
result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val);
return GSL_SUCCESS;
}
else if(x < 6.0) {
result->val = 0.5 * x * log((x+1.0)/(x-1.0)) - 1.0;
result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val);
return GSL_SUCCESS;
}
else if(x*GSL_SQRT_DBL_MIN < 0.99/M_SQRT3) {
const double y = 1/(x*x);
const double c1 = 3.0/5.0;
const double c2 = 3.0/7.0;
const double c3 = 3.0/9.0;
const double c4 = 3.0/11.0;
const double c5 = 3.0/13.0;
const double c6 = 3.0/15.0;
const double c7 = 3.0/17.0;
const double c8 = 3.0/19.0;
const double sum = 1.0 + y*(c1 + y*(c2 + y*(c3 + y*(c4 + y*(c5 + y*(c6 + y*(c7 + y*c8)))))));
result->val = sum / (3.0*x*x);
result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val);
return GSL_SUCCESS;
}
else {
UNDERFLOW_ERROR(result);
}
}
int
gsl_sf_legendre_Ql_e(const int l, const double x, gsl_sf_result * result)
{
/* CHECK_POINTER(result) */
if(x <= -1.0 || x == 1.0 || l < 0) {
DOMAIN_ERROR(result);
}
else if(l == 0) {
return gsl_sf_legendre_Q0_e(x, result);
}
else if(l == 1) {
return gsl_sf_legendre_Q1_e(x, result);
}
else if(l > 100000) {
return legendre_Ql_asymp_unif(l, x, result);
}
else if(x < 1.0){
/* Forward recurrence.
*/
gsl_sf_result Q0, Q1;
int stat_Q0 = gsl_sf_legendre_Q0_e(x, &Q0);
int stat_Q1 = gsl_sf_legendre_Q1_e(x, &Q1);
double Qellm1 = Q0.val;
double Qell = Q1.val;
double Qellp1;
int ell;
for(ell=1; ell<l; ell++) {
Qellp1 = (x*(2.0*ell + 1.0) * Qell - ell * Qellm1) / (ell + 1.0);
Qellm1 = Qell;
Qell = Qellp1;
}
result->val = Qell;
result->err = GSL_DBL_EPSILON * l * fabs(result->val);
return GSL_ERROR_SELECT_2(stat_Q0, stat_Q1);
}
else {
/* x > 1.0 */
double rat;
int stat_CF1 = legendreQ_CF1_xgt1(l, 0.0, 0.0, x, &rat);
int stat_Q;
double Qellp1 = rat * GSL_SQRT_DBL_MIN;
double Qell = GSL_SQRT_DBL_MIN;
double Qellm1;
int ell;
for(ell=l; ell>0; ell--) {
Qellm1 = (x * (2.0*ell + 1.0) * Qell - (ell+1.0) * Qellp1) / ell;
Qellp1 = Qell;
Qell = Qellm1;
}
if(fabs(Qell) > fabs(Qellp1)) {
gsl_sf_result Q0;
stat_Q = gsl_sf_legendre_Q0_e(x, &Q0);
result->val = GSL_SQRT_DBL_MIN * Q0.val / Qell;
result->err = l * GSL_DBL_EPSILON * fabs(result->val);
}
else {
gsl_sf_result Q1;
stat_Q = gsl_sf_legendre_Q1_e(x, &Q1);
result->val = GSL_SQRT_DBL_MIN * Q1.val / Qellp1;
result->err = l * GSL_DBL_EPSILON * fabs(result->val);
}
return GSL_ERROR_SELECT_2(stat_Q, stat_CF1);
}
}
/*-*-*-*-*-*-*-*-*-* Functions w/ Natural Prototypes *-*-*-*-*-*-*-*-*-*-*/
#include "eval.h"
double gsl_sf_legendre_Q0(const double x)
{
EVAL_RESULT(gsl_sf_legendre_Q0_e(x, &result));
}
double gsl_sf_legendre_Q1(const double x)
{
EVAL_RESULT(gsl_sf_legendre_Q1_e(x, &result));
}
double gsl_sf_legendre_Ql(const int l, const double x)
{
EVAL_RESULT(gsl_sf_legendre_Ql_e(l, x, &result));
}