/* 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 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., 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;
}