summaryrefslogtreecommitdiff
path: root/gsl-1.9/specfunc/mathieu_coeff.c
diff options
context:
space:
mode:
Diffstat (limited to 'gsl-1.9/specfunc/mathieu_coeff.c')
-rw-r--r--gsl-1.9/specfunc/mathieu_coeff.c348
1 files changed, 348 insertions, 0 deletions
diff --git a/gsl-1.9/specfunc/mathieu_coeff.c b/gsl-1.9/specfunc/mathieu_coeff.c
new file mode 100644
index 0000000..960b37e
--- /dev/null
+++ b/gsl-1.9/specfunc/mathieu_coeff.c
@@ -0,0 +1,348 @@
+/* specfunc/mathieu_coeff.c
+ *
+ * Copyright (C) 2002 Lowell Johnson
+ *
+ * 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., 675 Mass Ave, Cambridge, MA 02139, USA.
+ */
+
+/* Author: L. Johnson */
+
+#include <config.h>
+#include <stdlib.h>
+#include <math.h>
+#include <gsl/gsl_sf_mathieu.h>
+
+
+/*****************************************************************************
+ * backward_recurse
+ *
+ * Purpose:
+ ****************************************************************************/
+static void backward_recurse_c(double aa, double qq, double xx, double *ff,
+ double *gx, int even_odd, int ni)
+{
+ int ii, nn;
+ double g1;
+
+
+ g1 = *gx;
+ ff[ni] = xx;
+
+ if (even_odd == 0)
+ {
+ for (ii=0; ii<ni; ii++)
+ {
+ nn = GSL_SF_MATHIEU_COEFF - ii - 1;
+ ff[ni-ii-1] = -1.0/((4*nn*nn - aa)/qq + ff[ni-ii]);
+ }
+ if (ni == GSL_SF_MATHIEU_COEFF - 1)
+ ff[0] *= 2.0;
+ }
+ else
+ {
+ for (ii=0; ii<ni; ii++)
+ {
+ nn = GSL_SF_MATHIEU_COEFF - ii - 1;
+ ff[ni-ii-1] = -1.0/(((2*nn + 1)*(2*nn + 1) - aa)/qq + ff[ni-ii]);
+ }
+ }
+
+ *gx = ff[0] - g1;
+}
+
+
+static void backward_recurse_s(double aa, double qq, double xx, double *ff,
+ double *gx, int even_odd, int ni)
+{
+ int ii, nn;
+ double g1;
+
+
+ g1 = *gx;
+ ff[ni] = xx;
+
+ if (even_odd == 0)
+ {
+ for (ii=0; ii<ni; ii++)
+ {
+ nn = GSL_SF_MATHIEU_COEFF - ii - 1;
+ ff[ni-ii-1] = -1.0/((4*(nn + 1)*(nn + 1) - aa)/qq + ff[ni-ii]);
+ }
+ }
+ else
+ {
+ for (ii=0; ii<ni; ii++)
+ {
+ nn = GSL_SF_MATHIEU_COEFF - ii - 1;
+ ff[ni-ii-1] = -1.0/(((2*nn + 1)*(2*nn + 1) - aa)/qq + ff[ni-ii]);
+ }
+ }
+
+ *gx = ff[0] - g1;
+}
+
+
+int gsl_sf_mathieu_a_coeff(int order, double qq, double aa, double coeff[])
+{
+ int ni, nn, ii, even_odd;
+ double eps, g1, g2, x1, x2, e1, e2, de, xh, sum, ratio,
+ ff[GSL_SF_MATHIEU_COEFF];
+
+
+ eps = 1e-14;
+ coeff[0] = 1.0;
+
+ even_odd = 0;
+ if (order % 2 != 0)
+ even_odd = 1;
+
+ /* If the coefficient array is not large enough to hold all necessary
+ coefficients, error out. */
+ if (order > GSL_SF_MATHIEU_COEFF)
+ return GSL_FAILURE;
+
+ /* Handle the trivial case where q = 0. */
+ if (qq == 0.0)
+ {
+ for (ii=0; ii<GSL_SF_MATHIEU_COEFF; ii++)
+ coeff[ii] = 0.0;
+
+ coeff[order/2] = 1.0;
+
+ return GSL_SUCCESS;
+ }
+
+ if (order < 5)
+ {
+ nn = 0;
+ sum = 0.0;
+ if (even_odd == 0)
+ ratio = aa/qq;
+ else
+ ratio = (aa - 1 - qq)/qq;
+ }
+ else
+ {
+ if (even_odd == 0)
+ {
+ coeff[1] = aa/qq;
+ coeff[2] = (aa - 4)/qq*coeff[1] - 2;
+ sum = coeff[0] + coeff[1] + coeff[2];
+ for (ii=3; ii<order/2+1; ii++)
+ {
+ coeff[ii] = (aa - 4*(ii - 1)*(ii - 1))/qq*coeff[ii-1] -
+ coeff[ii-2];
+ sum += coeff[ii];
+ }
+ }
+ else
+ {
+ coeff[1] = (aa - 1)/qq - 1;
+ sum = coeff[0] + coeff[1];
+ for (ii=2; ii<order/2+1; ii++)
+ {
+ coeff[ii] = (aa - (2*ii - 1)*(2*ii - 1))/qq*coeff[ii-1] -
+ coeff[ii-2];
+ sum += coeff[ii];
+ }
+ }
+
+ nn = ii - 1;
+
+ ratio = coeff[nn]/coeff[nn-1];
+ }
+
+ ni = GSL_SF_MATHIEU_COEFF - nn - 1;
+
+ /* Compute first two points to start root-finding. */
+ if (even_odd == 0)
+ x1 = -qq/(4.0*GSL_SF_MATHIEU_COEFF*GSL_SF_MATHIEU_COEFF);
+ else
+ x1 = -qq/((2.0*GSL_SF_MATHIEU_COEFF + 1.0)*(2.0*GSL_SF_MATHIEU_COEFF + 1.0));
+ g1 = ratio;
+ backward_recurse_c(aa, qq, x1, ff, &g1, even_odd, ni);
+ x2 = g1;
+ g2 = ratio;
+ backward_recurse_c(aa, qq, x2, ff, &g2, even_odd, ni);
+
+ /* Find the root. */
+ while (1)
+ {
+ /* Compute the relative error. */
+ e1 = g1 - x1;
+ e2 = g2 - x2;
+ de = e1 - e2;
+
+ /* If we are close enough to the root, break... */
+ if (fabs(de) < eps)
+ break;
+
+ /* Otherwise, determine the next guess and try again. */
+ xh = (e1*x2 - e2*x1)/de;
+ x1 = x2;
+ g1 = g2;
+ x2 = xh;
+ g2 = ratio;
+ backward_recurse_c(aa, qq, x2, ff, &g2, even_odd, ni);
+ }
+
+ /* Compute the rest of the coefficients. */
+ sum += coeff[nn];
+ for (ii=nn+1; ii<GSL_SF_MATHIEU_COEFF; ii++)
+ {
+ coeff[ii] = ff[ii-nn-1]*coeff[ii-1];
+ sum += coeff[ii];
+
+ /* If the coefficients are getting really small, set the remainder
+ to zero. */
+ if (fabs(coeff[ii]) < 1e-20)
+ {
+ for (; ii<GSL_SF_MATHIEU_COEFF;)
+ coeff[ii++] = 0.0;
+ }
+ }
+
+ /* Normalize the coefficients. */
+ for (ii=0; ii<GSL_SF_MATHIEU_COEFF; ii++)
+ coeff[ii] /= sum;
+
+ return GSL_SUCCESS;
+}
+
+
+int gsl_sf_mathieu_b_coeff(int order, double qq, double aa, double coeff[])
+{
+ int ni, nn, ii, even_odd;
+ double eps, g1, g2, x1, x2, e1, e2, de, xh, sum, ratio,
+ ff[GSL_SF_MATHIEU_COEFF];
+
+
+ eps = 1e-10;
+ coeff[0] = 1.0;
+
+ even_odd = 0;
+ if (order % 2 != 0)
+ even_odd = 1;
+
+ /* If the coefficient array is not large enough to hold all necessary
+ coefficients, error out. */
+ if (order > GSL_SF_MATHIEU_COEFF)
+ return GSL_FAILURE;
+
+ /* Handle the trivial case where q = 0. */
+ if (qq == 0.0)
+ {
+ for (ii=0; ii<GSL_SF_MATHIEU_COEFF; ii++)
+ coeff[ii] = 0.0;
+
+ coeff[(order-1)/2] = 1.0;
+
+ return GSL_SUCCESS;
+ }
+
+ if (order < 5)
+ {
+ nn = 0;
+ sum = 0.0;
+ if (even_odd == 0)
+ ratio = (aa - 4)/qq;
+ else
+ ratio = (aa - 1 - qq)/qq;
+ }
+ else
+ {
+ if (even_odd == 0)
+ {
+ coeff[1] = (aa - 4)/qq;
+ sum = 2*coeff[0] + 4*coeff[1];
+ for (ii=2; ii<order/2; ii++)
+ {
+ coeff[ii] = (aa - 4*ii*ii)/qq*coeff[ii-1] - coeff[ii-2];
+ sum += 2*(ii + 1)*coeff[ii];
+ }
+ }
+ else
+ {
+ coeff[1] = (aa - 1)/qq + 1;
+ sum = coeff[0] + 3*coeff[1];
+ for (ii=2; ii<order/2+1; ii++)
+ {
+ coeff[ii] = (aa - (2*ii - 1)*(2*ii - 1))/qq*coeff[ii-1] -
+ coeff[ii-2];
+ sum += (2*(ii + 1) - 1)*coeff[ii];
+ }
+ }
+
+ nn = ii - 1;
+
+ ratio = coeff[nn]/coeff[nn-1];
+ }
+
+ ni = GSL_SF_MATHIEU_COEFF - nn - 1;
+
+ /* Compute first two points to start root-finding. */
+ if (even_odd == 0)
+ x1 = -qq/(4.0*(GSL_SF_MATHIEU_COEFF + 1.0)*(GSL_SF_MATHIEU_COEFF + 1.0));
+ else
+ x1 = -qq/((2.0*GSL_SF_MATHIEU_COEFF + 1.0)*(2.0*GSL_SF_MATHIEU_COEFF + 1.0));
+ g1 = ratio;
+ backward_recurse_s(aa, qq, x1, ff, &g1, even_odd, ni);
+ x2 = g1;
+ g2 = ratio;
+ backward_recurse_s(aa, qq, x2, ff, &g2, even_odd, ni);
+
+ /* Find the root. */
+ while (1)
+ {
+ /* Compute the relative error. */
+ e1 = g1 - x1;
+ e2 = g2 - x2;
+ de = e1 - e2;
+
+ /* If we are close enough to the root, break... */
+ if (fabs(de) < eps)
+ break;
+
+ /* Otherwise, determine the next guess and try again. */
+ xh = (e1*x2 - e2*x1)/de;
+ x1 = x2;
+ g1 = g2;
+ x2 = xh;
+ g2 = ratio;
+ backward_recurse_s(aa, qq, x2, ff, &g2, even_odd, ni);
+ }
+
+ /* Compute the rest of the coefficients. */
+ sum += 2*(nn + 1)*coeff[nn];
+ for (ii=nn+1; ii<GSL_SF_MATHIEU_COEFF; ii++)
+ {
+ coeff[ii] = ff[ii-nn-1]*coeff[ii-1];
+ sum += 2*(ii + 1)*coeff[ii];
+
+ /* If the coefficients are getting really small, set the remainder
+ to zero. */
+ if (fabs(coeff[ii]) < 1e-20)
+ {
+ for (; ii<GSL_SF_MATHIEU_COEFF;)
+ coeff[ii++] = 0.0;
+ }
+ }
+
+ /* Normalize the coefficients. */
+ for (ii=0; ii<GSL_SF_MATHIEU_COEFF; ii++)
+ coeff[ii] /= sum;
+
+ return GSL_SUCCESS;
+}
generated by cgit v1.2.3 (git 2.25.1) at 2024-05-01 07:57:22 +0000