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Diffstat (limited to 'gsl-1.9/specfunc/laguerre.c')
| -rw-r--r-- | gsl-1.9/specfunc/laguerre.c | 334 |
1 files changed, 334 insertions, 0 deletions
diff --git a/gsl-1.9/specfunc/laguerre.c b/gsl-1.9/specfunc/laguerre.c new file mode 100644 index 0000000..355b35e --- /dev/null +++ b/gsl-1.9/specfunc/laguerre.c @@ -0,0 +1,334 @@ +/* specfunc/laguerre.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 2 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_exp.h> +#include <gsl/gsl_sf_gamma.h> +#include <gsl/gsl_sf_laguerre.h> + +#include "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 "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)); +} |