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+/* integration/qmomof.c
+ *
+ * Copyright (C) 1996, 1997, 1998, 1999, 2000 Brian Gough
+ *
+ * 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.
+ */
+
+#include <config.h>
+#include <stdlib.h>
+#include <gsl/gsl_integration.h>
+#include <gsl/gsl_errno.h>
+
+static void
+compute_moments (double par, double * cheb);
+
+static int
+dgtsl (size_t n, double *c, double *d, double *e, double *b);
+
+gsl_integration_qawo_table *
+gsl_integration_qawo_table_alloc (double omega, double L,
+ enum gsl_integration_qawo_enum sine,
+ size_t n)
+{
+ gsl_integration_qawo_table *t;
+ double * chebmo;
+
+ if (n == 0)
+ {
+ GSL_ERROR_VAL ("table length n must be positive integer",
+ GSL_EDOM, 0);
+ }
+
+ t = (gsl_integration_qawo_table *)
+ malloc (sizeof (gsl_integration_qawo_table));
+
+ if (t == 0)
+ {
+ GSL_ERROR_VAL ("failed to allocate space for qawo_table struct",
+ GSL_ENOMEM, 0);
+ }
+
+ chebmo = (double *) malloc (25 * n * sizeof (double));
+
+ if (chebmo == 0)
+ {
+ free (t);
+ GSL_ERROR_VAL ("failed to allocate space for chebmo block",
+ GSL_ENOMEM, 0);
+ }
+
+ t->n = n;
+ t->sine = sine;
+ t->omega = omega;
+ t->L = L;
+ t->par = 0.5 * omega * L;
+ t->chebmo = chebmo;
+
+ /* precompute the moments */
+
+ {
+ size_t i;
+ double scale = 1.0;
+
+ for (i = 0 ; i < t->n; i++)
+ {
+ compute_moments (t->par * scale, t->chebmo + 25*i);
+ scale *= 0.5;
+ }
+ }
+
+ return t;
+}
+
+int
+gsl_integration_qawo_table_set (gsl_integration_qawo_table * t,
+ double omega, double L,
+ enum gsl_integration_qawo_enum sine)
+{
+ t->omega = omega;
+ t->sine = sine;
+ t->L = L;
+ t->par = 0.5 * omega * L;
+
+ /* recompute the moments */
+
+ {
+ size_t i;
+ double scale = 1.0;
+
+ for (i = 0 ; i < t->n; i++)
+ {
+ compute_moments (t->par * scale, t->chebmo + 25*i);
+ scale *= 0.5;
+ }
+ }
+
+ return GSL_SUCCESS;
+}
+
+
+int
+gsl_integration_qawo_table_set_length (gsl_integration_qawo_table * t,
+ double L)
+{
+ /* return immediately if the length is the same as the old length */
+
+ if (L == t->L)
+ return GSL_SUCCESS;
+
+ /* otherwise reset the table and compute the new parameters */
+
+ t->L = L;
+ t->par = 0.5 * t->omega * L;
+
+ /* recompute the moments */
+
+ {
+ size_t i;
+ double scale = 1.0;
+
+ for (i = 0 ; i < t->n; i++)
+ {
+ compute_moments (t->par * scale, t->chebmo + 25*i);
+ scale *= 0.5;
+ }
+ }
+
+ return GSL_SUCCESS;
+}
+
+
+void
+gsl_integration_qawo_table_free (gsl_integration_qawo_table * t)
+{
+ free (t->chebmo);
+ free (t);
+}
+
+static void
+compute_moments (double par, double *chebmo)
+{
+ double v[28], d[25], d1[25], d2[25];
+
+ const size_t noeq = 25;
+
+ const double par2 = par * par;
+ const double par4 = par2 * par2;
+ const double par22 = par2 + 2.0;
+
+ const double sinpar = sin (par);
+ const double cospar = cos (par);
+
+ size_t i;
+
+ /* compute the chebyschev moments with respect to cosine */
+
+ double ac = 8 * cospar;
+ double as = 24 * par * sinpar;
+
+ v[0] = 2 * sinpar / par;
+ v[1] = (8 * cospar + (2 * par2 - 8) * sinpar / par) / par2;
+ v[2] = (32 * (par2 - 12) * cospar
+ + (2 * ((par2 - 80) * par2 + 192) * sinpar) / par) / par4;
+
+ if (fabs (par) <= 24)
+ {
+ /* compute the moments as the solution of a boundary value
+ problem using the asyptotic expansion as an endpoint */
+
+ double an2, ass, asap;
+ double an = 6;
+ size_t k;
+
+ for (k = 0; k < noeq - 1; k++)
+ {
+ an2 = an * an;
+ d[k] = -2 * (an2 - 4) * (par22 - 2 * an2);
+ d2[k] = (an - 1) * (an - 2) * par2;
+ d1[k + 1] = (an + 3) * (an + 4) * par2;
+ v[k + 3] = as - (an2 - 4) * ac;
+ an = an + 2.0;
+ }
+
+ an2 = an * an;
+
+ d[noeq - 1] = -2 * (an2 - 4) * (par22 - 2 * an2);
+ v[noeq + 2] = as - (an2 - 4) * ac;
+ v[3] = v[3] - 56 * par2 * v[2];
+
+ ass = par * sinpar;
+ asap = (((((210 * par2 - 1) * cospar - (105 * par2 - 63) * ass) / an2
+ - (1 - 15 * par2) * cospar + 15 * ass) / an2
+ - cospar + 3 * ass) / an2
+ - cospar) / an2;
+ v[noeq + 2] = v[noeq + 2] - 2 * asap * par2 * (an - 1) * (an - 2);
+
+ dgtsl (noeq, d1, d, d2, v + 3);
+
+ }
+ else
+ {
+ /* compute the moments by forward recursion */
+ size_t k;
+ double an = 4;
+
+ for (k = 3; k < 13; k++)
+ {
+ double an2 = an * an;
+ v[k] = ((an2 - 4) * (2 * (par22 - 2 * an2) * v[k - 1] - ac)
+ + as - par2 * (an + 1) * (an + 2) * v[k - 2])
+ / (par2 * (an - 1) * (an - 2));
+ an = an + 2.0;
+ }
+ }
+
+
+ for (i = 0; i < 13; i++)
+ {
+ chebmo[2 * i] = v[i];
+ }
+
+ /* compute the chebyschev moments with respect to sine */
+
+ v[0] = 2 * (sinpar - par * cospar) / par2;
+ v[1] = (18 - 48 / par2) * sinpar / par2 + (-2 + 48 / par2) * cospar / par;
+
+ ac = -24 * par * cospar;
+ as = -8 * sinpar;
+
+ if (fabs (par) <= 24)
+ {
+ /* compute the moments as the solution of a boundary value
+ problem using the asyptotic expansion as an endpoint */
+
+ size_t k;
+ double an2, ass, asap;
+ double an = 5;
+
+ for (k = 0; k < noeq - 1; k++)
+ {
+ an2 = an * an;
+ d[k] = -2 * (an2 - 4) * (par22 - 2 * an2);
+ d2[k] = (an - 1) * (an - 2) * par2;
+ d1[k + 1] = (an + 3) * (an + 4) * par2;
+ v[k + 2] = ac + (an2 - 4) * as;
+ an = an + 2.0;
+ }
+
+ an2 = an * an;
+
+ d[noeq - 1] = -2 * (an2 - 4) * (par22 - 2 * an2);
+ v[noeq + 1] = ac + (an2 - 4) * as;
+ v[2] = v[2] - 42 * par2 * v[1];
+
+ ass = par * cospar;
+ asap = (((((105 * par2 - 63) * ass - (210 * par2 - 1) * sinpar) / an2
+ + (15 * par2 - 1) * sinpar
+ - 15 * ass) / an2 - sinpar - 3 * ass) / an2 - sinpar) / an2;
+ v[noeq + 1] = v[noeq + 1] - 2 * asap * par2 * (an - 1) * (an - 2);
+
+ dgtsl (noeq, d1, d, d2, v + 2);
+
+ }
+ else
+ {
+ /* compute the moments by forward recursion */
+ size_t k;
+ double an = 3;
+ for (k = 2; k < 12; k++)
+ {
+ double an2 = an * an;
+ v[k] = ((an2 - 4) * (2 * (par22 - 2 * an2) * v[k - 1] + as)
+ + ac - par2 * (an + 1) * (an + 2) * v[k - 2])
+ / (par2 * (an - 1) * (an - 2));
+ an = an + 2.0;
+ }
+ }
+
+ for (i = 0; i < 12; i++)
+ {
+ chebmo[2 * i + 1] = v[i];
+ }
+
+}
+
+static int
+dgtsl (size_t n, double *c, double *d, double *e, double *b)
+{
+ /* solves a tridiagonal matrix A x = b
+
+ c[1 .. n - 1] subdiagonal of the matrix A
+ d[0 .. n - 1] diagonal of the matrix A
+ e[0 .. n - 2] superdiagonal of the matrix A
+
+ b[0 .. n - 1] right hand side, replaced by the solution vector x */
+
+ size_t k;
+
+ c[0] = d[0];
+
+ if (n == 0)
+ {
+ return GSL_SUCCESS;
+ }
+
+ if (n == 1)
+ {
+ b[0] = b[0] / d[0] ;
+ return GSL_SUCCESS;
+ }
+
+ d[0] = e[0];
+ e[0] = 0;
+ e[n - 1] = 0;
+
+ for (k = 0; k < n - 1; k++)
+ {
+ size_t k1 = k + 1;
+
+ if (fabs (c[k1]) >= fabs (c[k]))
+ {
+ {
+ double t = c[k1];
+ c[k1] = c[k];
+ c[k] = t;
+ };
+ {
+ double t = d[k1];
+ d[k1] = d[k];
+ d[k] = t;
+ };
+ {
+ double t = e[k1];
+ e[k1] = e[k];
+ e[k] = t;
+ };
+ {
+ double t = b[k1];
+ b[k1] = b[k];
+ b[k] = t;
+ };
+ }
+
+ if (c[k] == 0)
+ {
+ return GSL_FAILURE ;
+ }
+
+ {
+ double t = -c[k1] / c[k];
+
+ c[k1] = d[k1] + t * d[k];
+ d[k1] = e[k1] + t * e[k];
+ e[k1] = 0;
+ b[k1] = b[k1] + t * b[k];
+ }
+
+ }
+
+ if (c[n - 1] == 0)
+ {
+ return GSL_FAILURE;
+ }
+
+
+ b[n - 1] = b[n - 1] / c[n - 1];
+
+ b[n - 2] = (b[n - 2] - d[n - 2] * b[n - 1]) / c[n - 2];
+
+ for (k = n ; k > 2; k--)
+ {
+ size_t kb = k - 3;
+ b[kb] = (b[kb] - d[kb] * b[kb + 1] - e[kb] * b[kb + 2]) / c[kb];
+ }
+
+ return GSL_SUCCESS;
+}