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- /* specfunc/laguerre.c
- *
- * Copyright (C) 2007 Brian Gough
- * 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 "gsl__config.h"
- #include "gsl_math.h"
- #include "gsl_errno.h"
- #include "gsl_sf_exp.h"
- #include "gsl_sf_gamma.h"
- #include "gsl_sf_laguerre.h"
- #include "gsl_specfunc__error.h"
- /*-*-*-*-*-*-*-*-*-*-*-* Private Section *-*-*-*-*-*-*-*-*-*-*-*/
- /* based on the large 2b-4a asymptotic for 1F1
- * [Abramowitz+Stegun, 13.5.21]
- * L^a_n(x) = (a+1)_n / n! 1F1(-n,a+1,x)
- *
- * The second term (ser_term2) is from Slater,"The Confluent
- * Hypergeometric Function" p.73. I think there may be an error in
- * the first term of the expression given there, comparing with AS
- * 13.5.21 (cf sin(a\pi+\Theta) vs sin(a\pi) + sin(\Theta)) - but the
- * second term appears correct.
- *
- */
- static
- int
- laguerre_large_n(const int n, const double alpha, const double x,
- gsl_sf_result * result)
- {
- const double a = -n;
- const double b = alpha + 1.0;
- const double eta = 2.0*b - 4.0*a;
- const double cos2th = x/eta;
- const double sin2th = 1.0 - cos2th;
- const double eps = asin(sqrt(cos2th)); /* theta = pi/2 - eps */
- const double pre_h = 0.25*M_PI*M_PI*eta*eta*cos2th*sin2th;
- gsl_sf_result lg_b;
- gsl_sf_result lnfact;
- int stat_lg = gsl_sf_lngamma_e(b+n, &lg_b);
- int stat_lf = gsl_sf_lnfact_e(n, &lnfact);
- double pre_term1 = 0.5*(1.0-b)*log(0.25*x*eta);
- double pre_term2 = 0.25*log(pre_h);
- double lnpre_val = lg_b.val - lnfact.val + 0.5*x + pre_term1 - pre_term2;
- double lnpre_err = lg_b.err + lnfact.err + GSL_DBL_EPSILON * (fabs(pre_term1)+fabs(pre_term2));
- double phi1 = 0.25*eta*(2*eps + sin(2.0*eps));
- double ser_term1 = -sin(phi1);
- double A1 = (1.0/12.0)*(5.0/(4.0*sin2th)+(3.0*b*b-6.0*b+2.0)*sin2th - 1.0);
- double ser_term2 = -A1 * cos(phi1)/(0.25*eta*sin(2.0*eps));
- double ser_val = ser_term1 + ser_term2;
- double ser_err = ser_term2*ser_term2 + GSL_DBL_EPSILON * (fabs(ser_term1) + fabs(ser_term2));
- int stat_e = gsl_sf_exp_mult_err_e(lnpre_val, lnpre_err, ser_val, ser_err, result);
- result->err += 2.0 * GSL_SQRT_DBL_EPSILON * fabs(result->val);
- return GSL_ERROR_SELECT_3(stat_e, stat_lf, stat_lg);
- }
- /* Evaluate polynomial based on confluent hypergeometric representation.
- *
- * L^a_n(x) = (a+1)_n / n! 1F1(-n,a+1,x)
- *
- * assumes n > 0 and a != negative integer greater than -n
- */
- static
- int
- laguerre_n_cp(const int n, const double a, const double x, gsl_sf_result * result)
- {
- gsl_sf_result lnfact;
- gsl_sf_result lg1;
- gsl_sf_result lg2;
- double s1, s2;
- int stat_f = gsl_sf_lnfact_e(n, &lnfact);
- int stat_g1 = gsl_sf_lngamma_sgn_e(a+1.0+n, &lg1, &s1);
- int stat_g2 = gsl_sf_lngamma_sgn_e(a+1.0, &lg2, &s2);
- double poly_1F1_val = 1.0;
- double poly_1F1_err = 0.0;
- int stat_e;
- int k;
- double lnpre_val = (lg1.val - lg2.val) - lnfact.val;
- double lnpre_err = lg1.err + lg2.err + lnfact.err + 2.0 * GSL_DBL_EPSILON * fabs(lnpre_val);
- for(k=n-1; k>=0; k--) {
- double t = (-n+k)/(a+1.0+k) * (x/(k+1));
- double r = t + 1.0/poly_1F1_val;
- if(r > 0.9*GSL_DBL_MAX/poly_1F1_val) {
- /* internal error only, don't call the error handler */
- INTERNAL_OVERFLOW_ERROR(result);
- }
- else {
- /* Collect the Horner terms. */
- poly_1F1_val = 1.0 + t * poly_1F1_val;
- poly_1F1_err += GSL_DBL_EPSILON + fabs(t) * poly_1F1_err;
- }
- }
- stat_e = gsl_sf_exp_mult_err_e(lnpre_val, lnpre_err,
- poly_1F1_val, poly_1F1_err,
- result);
- return GSL_ERROR_SELECT_4(stat_e, stat_f, stat_g1, stat_g2);
- }
- /* Evaluate the polynomial based on the confluent hypergeometric
- * function in a safe way, with no restriction on the arguments.
- *
- * assumes x != 0
- */
- static
- int
- laguerre_n_poly_safe(const int n, const double a, const double x, gsl_sf_result * result)
- {
- const double b = a + 1.0;
- const double mx = -x;
- const double tc_sgn = (x < 0.0 ? 1.0 : (GSL_IS_ODD(n) ? -1.0 : 1.0));
- gsl_sf_result tc;
- int stat_tc = gsl_sf_taylorcoeff_e(n, fabs(x), &tc);
- if(stat_tc == GSL_SUCCESS) {
- double term = tc.val * tc_sgn;
- double sum_val = term;
- double sum_err = tc.err;
- int k;
- for(k=n-1; k>=0; k--) {
- term *= ((b+k)/(n-k))*(k+1.0)/mx;
- sum_val += term;
- sum_err += 4.0 * GSL_DBL_EPSILON * fabs(term);
- }
- result->val = sum_val;
- result->err = sum_err + 2.0 * GSL_DBL_EPSILON * fabs(result->val);
- return GSL_SUCCESS;
- }
- else if(stat_tc == GSL_EOVRFLW) {
- result->val = 0.0; /* FIXME: should be Inf */
- result->err = 0.0;
- return stat_tc;
- }
- else {
- result->val = 0.0;
- result->err = 0.0;
- return stat_tc;
- }
- }
- /*-*-*-*-*-*-*-*-*-*-*-* Functions with Error Codes *-*-*-*-*-*-*-*-*-*-*/
- int
- gsl_sf_laguerre_1_e(const double a, const double x, gsl_sf_result * result)
- {
- /* CHECK_POINTER(result) */
- {
- result->val = 1.0 + a - x;
- result->err = 2.0 * GSL_DBL_EPSILON * (1.0 + fabs(a) + fabs(x));
- return GSL_SUCCESS;
- }
- }
- int
- gsl_sf_laguerre_2_e(const double a, const double x, gsl_sf_result * result)
- {
- /* CHECK_POINTER(result) */
- if(a == -2.0) {
- result->val = 0.5*x*x;
- result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val);
- return GSL_SUCCESS;
- }
- else {
- double c0 = 0.5 * (2.0+a)*(1.0+a);
- double c1 = -(2.0+a);
- double c2 = -0.5/(2.0+a);
- result->val = c0 + c1*x*(1.0 + c2*x);
- result->err = 2.0 * GSL_DBL_EPSILON * (fabs(c0) + 2.0 * fabs(c1*x) * (1.0 + 2.0 * fabs(c2*x)));
- result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
- return GSL_SUCCESS;
- }
- }
- int
- gsl_sf_laguerre_3_e(const double a, const double x, gsl_sf_result * result)
- {
- /* CHECK_POINTER(result) */
- if(a == -2.0) {
- double x2_6 = x*x/6.0;
- result->val = x2_6 * (3.0 - x);
- result->err = x2_6 * (3.0 + fabs(x)) * 2.0 * GSL_DBL_EPSILON;
- result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
- return GSL_SUCCESS;
- }
- else if(a == -3.0) {
- result->val = -x*x/6.0;
- result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val);
- return GSL_SUCCESS;
- }
- else {
- double c0 = (3.0+a)*(2.0+a)*(1.0+a) / 6.0;
- double c1 = -c0 * 3.0 / (1.0+a);
- double c2 = -1.0/(2.0+a);
- double c3 = -1.0/(3.0*(3.0+a));
- result->val = c0 + c1*x*(1.0 + c2*x*(1.0 + c3*x));
- result->err = 1.0 + 2.0 * fabs(c3*x);
- result->err = 1.0 + 2.0 * fabs(c2*x) * result->err;
- result->err = 2.0 * GSL_DBL_EPSILON * (fabs(c0) + 2.0 * fabs(c1*x) * result->err);
- result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
- return GSL_SUCCESS;
- }
- }
- int gsl_sf_laguerre_n_e(const int n, const double a, const double x,
- gsl_sf_result * result)
- {
- /* CHECK_POINTER(result) */
- if(n < 0) {
- DOMAIN_ERROR(result);
- }
- else if(n == 0) {
- result->val = 1.0;
- result->err = 0.0;
- return GSL_SUCCESS;
- }
- else if(n == 1) {
- result->val = 1.0 + a - x;
- result->err = 2.0 * GSL_DBL_EPSILON * (1.0 + fabs(a) + fabs(x));
- return GSL_SUCCESS;
- }
- else if(x == 0.0) {
- double product = a + 1.0;
- int k;
- for(k=2; k<=n; k++) {
- product *= (a + k)/k;
- }
- result->val = product;
- result->err = 2.0 * (n + 1.0) * GSL_DBL_EPSILON * fabs(product) + GSL_DBL_EPSILON;
- return GSL_SUCCESS;
- }
- else if(x < 0.0 && a > -1.0) {
- /* In this case all the terms in the polynomial
- * are of the same sign. Note that this also
- * catches overflows correctly.
- */
- return laguerre_n_cp(n, a, x, result);
- }
- else if(n < 5 || (x > 0.0 && a < -n-1)) {
- /* Either the polynomial will not lose too much accuracy
- * or all the terms are negative. In any case,
- * the error estimate here is good. We try both
- * explicit summation methods, as they have different
- * characteristics. One may underflow/overflow while the
- * other does not.
- */
- if(laguerre_n_cp(n, a, x, result) == GSL_SUCCESS)
- return GSL_SUCCESS;
- else
- return laguerre_n_poly_safe(n, a, x, result);
- }
- else if(n > 1.0e+07 && x > 0.0 && a > -1.0 && x < 2.0*(a+1.0)+4.0*n) {
- return laguerre_large_n(n, a, x, result);
- }
- else if(a >= 0.0 || (x > 0.0 && a < -n-1)) {
- gsl_sf_result lg2;
- int stat_lg2 = gsl_sf_laguerre_2_e(a, x, &lg2);
- double Lkm1 = 1.0 + a - x;
- double Lk = lg2.val;
- double Lkp1;
- int k;
- for(k=2; k<n; k++) {
- Lkp1 = (-(k+a)*Lkm1 + (2.0*k+a+1.0-x)*Lk)/(k+1.0);
- Lkm1 = Lk;
- Lk = Lkp1;
- }
- result->val = Lk;
- result->err = (fabs(lg2.err/lg2.val) + GSL_DBL_EPSILON) * n * fabs(Lk);
- return stat_lg2;
- }
- else {
- /* Despair... or magic? */
- return laguerre_n_poly_safe(n, a, x, result);
- }
- }
- /*-*-*-*-*-*-*-*-*-* Functions w/ Natural Prototypes *-*-*-*-*-*-*-*-*-*-*/
- #include "gsl_specfunc__eval.h"
- double gsl_sf_laguerre_1(double a, double x)
- {
- EVAL_RESULT(gsl_sf_laguerre_1_e(a, x, &result));
- }
- double gsl_sf_laguerre_2(double a, double x)
- {
- EVAL_RESULT(gsl_sf_laguerre_2_e(a, x, &result));
- }
- double gsl_sf_laguerre_3(double a, double x)
- {
- EVAL_RESULT(gsl_sf_laguerre_3_e(a, x, &result));
- }
- double gsl_sf_laguerre_n(int n, double a, double x)
- {
- EVAL_RESULT(gsl_sf_laguerre_n_e(n, a, x, &result));
- }
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