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+/* specfunc/bessel_Kn.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_gamma.h>
+#include <gsl/gsl_sf_psi.h>
+#include <gsl/gsl_sf_bessel.h>
+
+#include "error.h"
+
+#include "bessel.h"
+
+/*-*-*-*-*-*-*-*-*-*-*-* Private Section *-*-*-*-*-*-*-*-*-*-*-*/
+
+/* [Abramowitz+Stegun, 9.6.11]
+ * assumes n >= 1
+ */
+static
+int
+bessel_Kn_scaled_small_x(const int n, const double x, gsl_sf_result * result)
+{
+ int k;
+ double y = 0.25 * x * x;
+ double ln_x_2 = log(0.5*x);
+ double ex = exp(x);
+ gsl_sf_result ln_nm1_fact;
+ double k_term;
+ double term1, sum1, ln_pre1;
+ double term2, sum2, pre2;
+
+ gsl_sf_lnfact_e((unsigned int)(n-1), &ln_nm1_fact);
+
+ ln_pre1 = -n*ln_x_2 + ln_nm1_fact.val;
+ if(ln_pre1 > GSL_LOG_DBL_MAX - 3.0) GSL_ERROR ("error", GSL_EOVRFLW);
+
+ sum1 = 1.0;
+ k_term = 1.0;
+ for(k=1; k<=n-1; k++) {
+ k_term *= -y/(k * (n-k));
+ sum1 += k_term;
+ }
+ term1 = 0.5 * exp(ln_pre1) * sum1;
+
+ pre2 = 0.5 * exp(n*ln_x_2);
+ if(pre2 > 0.0) {
+ const int KMAX = 20;
+ gsl_sf_result psi_n;
+ gsl_sf_result npk_fact;
+ double yk = 1.0;
+ double k_fact = 1.0;
+ double psi_kp1 = -M_EULER;
+ double psi_npkp1;
+ gsl_sf_psi_int_e(n, &psi_n);
+ gsl_sf_fact_e((unsigned int)n, &npk_fact);
+ psi_npkp1 = psi_n.val + 1.0/n;
+ sum2 = (psi_kp1 + psi_npkp1 - 2.0*ln_x_2)/npk_fact.val;
+ for(k=1; k<KMAX; k++) {
+ psi_kp1 += 1.0/k;
+ psi_npkp1 += 1.0/(n+k);
+ k_fact *= k;
+ npk_fact.val *= n+k;
+ yk *= y;
+ k_term = yk*(psi_kp1 + psi_npkp1 - 2.0*ln_x_2)/(k_fact*npk_fact.val);
+ sum2 += k_term;
+ }
+ term2 = ( GSL_IS_ODD(n) ? -1.0 : 1.0 ) * pre2 * sum2;
+ }
+ else {
+ term2 = 0.0;
+ }
+
+ result->val = ex * (term1 + term2);
+ result->err = ex * GSL_DBL_EPSILON * (fabs(ln_pre1)*fabs(term1) + fabs(term2));
+ result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
+
+ return GSL_SUCCESS;
+}
+
+
+/*-*-*-*-*-*-*-*-*-*-*-* Functions with Error Codes *-*-*-*-*-*-*-*-*-*-*-*/
+
+int gsl_sf_bessel_Kn_scaled_e(int n, const double x, gsl_sf_result * result)
+{
+ n = abs(n); /* K(-n, z) = K(n, z) */
+
+ /* CHECK_POINTER(result) */
+
+ if(x <= 0.0) {
+ DOMAIN_ERROR(result);
+ }
+ else if(n == 0) {
+ return gsl_sf_bessel_K0_scaled_e(x, result);
+ }
+ else if(n == 1) {
+ return gsl_sf_bessel_K1_scaled_e(x, result);
+ }
+ else if(x <= 5.0) {
+ return bessel_Kn_scaled_small_x(n, x, result);
+ }
+ else if(GSL_ROOT3_DBL_EPSILON * x > 0.25 * (n*n + 1)) {
+ return gsl_sf_bessel_Knu_scaled_asympx_e((double)n, x, result);
+ }
+ else if(GSL_MIN(0.29/(n*n), 0.5/(n*n + x*x)) < GSL_ROOT3_DBL_EPSILON) {
+ return gsl_sf_bessel_Knu_scaled_asymp_unif_e((double)n, x, result);
+ }
+ else {
+ /* Upward recurrence. [Gradshteyn + Ryzhik, 8.471.1] */
+ double two_over_x = 2.0/x;
+ gsl_sf_result r_b_jm1;
+ gsl_sf_result r_b_j;
+ int stat_0 = gsl_sf_bessel_K0_scaled_e(x, &r_b_jm1);
+ int stat_1 = gsl_sf_bessel_K1_scaled_e(x, &r_b_j);
+ double b_jm1 = r_b_jm1.val;
+ double b_j = r_b_j.val;
+ double b_jp1;
+ int j;
+
+ for(j=1; j<n; j++) {
+ b_jp1 = b_jm1 + j * two_over_x * b_j;
+ b_jm1 = b_j;
+ b_j = b_jp1;
+ }
+
+ result->val = b_j;
+ result->err = n * (fabs(b_j) * (fabs(r_b_jm1.err/r_b_jm1.val) + fabs(r_b_j.err/r_b_j.val)));
+ result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
+
+ return GSL_ERROR_SELECT_2(stat_0, stat_1);
+ }
+}
+
+
+int gsl_sf_bessel_Kn_e(const int n, const double x, gsl_sf_result * result)
+{
+ const int status = gsl_sf_bessel_Kn_scaled_e(n, x, result);
+ const double ex = exp(-x);
+ result->val *= ex;
+ result->err *= ex;
+ result->err += x * GSL_DBL_EPSILON * fabs(result->val);
+ return status;
+}
+
+
+int gsl_sf_bessel_Kn_scaled_array(const int nmin, const int nmax, const double x, double * result_array)
+{
+ /* CHECK_POINTER(result_array) */
+
+ if(nmin < 0 || nmax < nmin || x <= 0.0) {
+ int j;
+ for(j=0; j<=nmax-nmin; j++) result_array[j] = 0.0;
+ GSL_ERROR ("domain error", GSL_EDOM);
+ }
+ else if(nmax == 0) {
+ gsl_sf_result b;
+ int stat = gsl_sf_bessel_K0_scaled_e(x, &b);
+ result_array[0] = b.val;
+ return stat;
+ }
+ else {
+ double two_over_x = 2.0/x;
+ gsl_sf_result r_Knm1;
+ gsl_sf_result r_Kn;
+ int stat_0 = gsl_sf_bessel_Kn_scaled_e(nmin, x, &r_Knm1);
+ int stat_1 = gsl_sf_bessel_Kn_scaled_e(nmin+1, x, &r_Kn);
+ int stat = GSL_ERROR_SELECT_2(stat_0, stat_1);
+ double Knp1;
+ double Kn = r_Kn.val;
+ double Knm1 = r_Knm1.val;
+ int n;
+
+ for(n=nmin+1; n<=nmax+1; n++) {
+ if(Knm1 < GSL_DBL_MAX) {
+ result_array[n-1-nmin] = Knm1;
+ Knp1 = Knm1 + n * two_over_x * Kn;
+ Knm1 = Kn;
+ Kn = Knp1;
+ }
+ else {
+ /* Overflow. Set the rest of the elements to
+ * zero and bug out.
+ * FIXME: Note: this relies on the convention
+ * that the test x < DBL_MIN fails for x not
+ * a number. This may be only an IEEE convention,
+ * so the portability is unclear.
+ */
+ int j;
+ for(j=n; j<=nmax+1; j++) result_array[j-1-nmin] = 0.0;
+ GSL_ERROR ("overflow", GSL_EOVRFLW);
+ }
+ }
+
+ return stat;
+ }
+}
+
+
+int
+gsl_sf_bessel_Kn_array(const int nmin, const int nmax, const double x, double * result_array)
+{
+ int status = gsl_sf_bessel_Kn_scaled_array(nmin, nmax, x, result_array);
+ double ex = exp(-x);
+ int i;
+ for(i=0; i<=nmax-nmin; i++) result_array[i] *= ex;
+ return status;
+}
+
+
+/*-*-*-*-*-*-*-*-*-* Functions w/ Natural Prototypes *-*-*-*-*-*-*-*-*-*-*/
+
+#include "eval.h"
+
+double gsl_sf_bessel_Kn_scaled(const int n, const double x)
+{
+ EVAL_RESULT(gsl_sf_bessel_Kn_scaled_e(n, x, &result));
+}
+
+double gsl_sf_bessel_Kn(const int n, const double x)
+{
+ EVAL_RESULT(gsl_sf_bessel_Kn_e(n, x, &result));
+}
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