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+/* interpolation/cspline.c
+ *
+ * Copyright (C) 1996, 1997, 1998, 1999, 2000, 2001, 2002, 2004 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 <stdlib.h>
+#include <gsl/gsl_errno.h>
+#include <gsl/gsl_linalg.h>
+#include <gsl/gsl_vector.h>
+#include "integ_eval.h"
+#include <gsl/gsl_interp.h>
+
+typedef struct
+{
+ double * c;
+ double * g;
+ double * diag;
+ double * offdiag;
+} cspline_state_t;
+
+
+/* common initialization */
+static void *
+cspline_alloc (size_t size)
+{
+ cspline_state_t * state = (cspline_state_t *) malloc (sizeof (cspline_state_t));
+
+ if (state == NULL)
+ {
+ GSL_ERROR_NULL("failed to allocate space for state", GSL_ENOMEM);
+ }
+
+ state->c = (double *) malloc (size * sizeof (double));
+
+ if (state->c == NULL)
+ {
+ free (state);
+ GSL_ERROR_NULL("failed to allocate space for c", GSL_ENOMEM);
+ }
+
+ state->g = (double *) malloc (size * sizeof (double));
+
+ if (state->g == NULL)
+ {
+ free (state->c);
+ free (state);
+ GSL_ERROR_NULL("failed to allocate space for g", GSL_ENOMEM);
+ }
+
+ state->diag = (double *) malloc (size * sizeof (double));
+
+ if (state->diag == NULL)
+ {
+ free (state->g);
+ free (state->c);
+ free (state);
+ GSL_ERROR_NULL("failed to allocate space for diag", GSL_ENOMEM);
+ }
+
+ state->offdiag = (double *) malloc (size * sizeof (double));
+
+ if (state->offdiag == NULL)
+ {
+ free (state->diag);
+ free (state->g);
+ free (state->c);
+ free (state);
+ GSL_ERROR_NULL("failed to allocate space for offdiag", GSL_ENOMEM);
+ }
+
+ return state;
+}
+
+
+/* natural spline calculation
+ * see [Engeln-Mullges + Uhlig, p. 254]
+ */
+static int
+cspline_init (void * vstate, const double xa[], const double ya[],
+ size_t size)
+{
+ cspline_state_t *state = (cspline_state_t *) vstate;
+
+ size_t i;
+ size_t num_points = size;
+ size_t max_index = num_points - 1; /* Engeln-Mullges + Uhlig "n" */
+ size_t sys_size = max_index - 1; /* linear system is sys_size x sys_size */
+
+ state->c[0] = 0.0;
+ state->c[max_index] = 0.0;
+
+ for (i = 0; i < sys_size; i++)
+ {
+ const double h_i = xa[i + 1] - xa[i];
+ const double h_ip1 = xa[i + 2] - xa[i + 1];
+ const double ydiff_i = ya[i + 1] - ya[i];
+ const double ydiff_ip1 = ya[i + 2] - ya[i + 1];
+ const double g_i = (h_i != 0.0) ? 1.0 / h_i : 0.0;
+ const double g_ip1 = (h_ip1 != 0.0) ? 1.0 / h_ip1 : 0.0;
+ state->offdiag[i] = h_ip1;
+ state->diag[i] = 2.0 * (h_ip1 + h_i);
+ state->g[i] = 3.0 * (ydiff_ip1 * g_ip1 - ydiff_i * g_i);
+ }
+
+ if (sys_size == 1)
+ {
+ state->c[1] = state->g[0] / state->diag[0];
+ return GSL_SUCCESS;
+ }
+ else
+ {
+ gsl_vector_view g_vec = gsl_vector_view_array(state->g, sys_size);
+ gsl_vector_view diag_vec = gsl_vector_view_array(state->diag, sys_size);
+ gsl_vector_view offdiag_vec = gsl_vector_view_array(state->offdiag, sys_size - 1);
+ gsl_vector_view solution_vec = gsl_vector_view_array ((state->c) + 1, sys_size);
+
+ int status = gsl_linalg_solve_symm_tridiag(&diag_vec.vector,
+ &offdiag_vec.vector,
+ &g_vec.vector,
+ &solution_vec.vector);
+ return status;
+ }
+}
+
+
+/* periodic spline calculation
+ * see [Engeln-Mullges + Uhlig, p. 256]
+ */
+static int
+cspline_init_periodic (void * vstate, const double xa[], const double ya[],
+ size_t size)
+{
+ cspline_state_t *state = (cspline_state_t *) vstate;
+
+ size_t i;
+ size_t num_points = size;
+ size_t max_index = num_points - 1; /* Engeln-Mullges + Uhlig "n" */
+ size_t sys_size = max_index; /* linear system is sys_size x sys_size */
+
+ if (sys_size == 2) {
+ /* solve 2x2 system */
+
+ const double h0 = xa[1] - xa[0];
+ const double h1 = xa[2] - xa[1];
+
+ const double A = 2.0*(h0 + h1);
+ const double B = h0 + h1;
+ double g[2];
+ double det;
+
+ g[0] = 3.0 * ((ya[2] - ya[1]) / h1 - (ya[1] - ya[0]) / h0);
+ g[1] = 3.0 * ((ya[1] - ya[2]) / h0 - (ya[2] - ya[1]) / h1);
+
+ det = 3.0 * (h0 + h1) * (h0 + h1);
+ state->c[1] = ( A * g[0] - B * g[1])/det;
+ state->c[2] = (-B * g[0] + A * g[1])/det;
+ state->c[0] = state->c[2];
+
+ return GSL_SUCCESS;
+ } else {
+
+ for (i = 0; i < sys_size-1; i++) {
+ const double h_i = xa[i + 1] - xa[i];
+ const double h_ip1 = xa[i + 2] - xa[i + 1];
+ const double ydiff_i = ya[i + 1] - ya[i];
+ const double ydiff_ip1 = ya[i + 2] - ya[i + 1];
+ const double g_i = (h_i != 0.0) ? 1.0 / h_i : 0.0;
+ const double g_ip1 = (h_ip1 != 0.0) ? 1.0 / h_ip1 : 0.0;
+ state->offdiag[i] = h_ip1;
+ state->diag[i] = 2.0 * (h_ip1 + h_i);
+ state->g[i] = 3.0 * (ydiff_ip1 * g_ip1 - ydiff_i * g_i);
+ }
+
+ i = sys_size - 1;
+
+ {
+ const double h_i = xa[i + 1] - xa[i];
+ const double h_ip1 = xa[1] - xa[0];
+ const double ydiff_i = ya[i + 1] - ya[i];
+ const double ydiff_ip1 = ya[1] - ya[0];
+ const double g_i = (h_i != 0.0) ? 1.0 / h_i : 0.0;
+ const double g_ip1 = (h_ip1 != 0.0) ? 1.0 / h_ip1 : 0.0;
+ state->offdiag[i] = h_ip1;
+ state->diag[i] = 2.0 * (h_ip1 + h_i);
+ state->g[i] = 3.0 * (ydiff_ip1 * g_ip1 - ydiff_i * g_i);
+ }
+
+ {
+ gsl_vector_view g_vec = gsl_vector_view_array(state->g, sys_size);
+ gsl_vector_view diag_vec = gsl_vector_view_array(state->diag, sys_size);
+ gsl_vector_view offdiag_vec = gsl_vector_view_array(state->offdiag, sys_size);
+ gsl_vector_view solution_vec = gsl_vector_view_array ((state->c) + 1, sys_size);
+
+ int status = gsl_linalg_solve_symm_cyc_tridiag(&diag_vec.vector,
+ &offdiag_vec.vector,
+ &g_vec.vector,
+ &solution_vec.vector);
+ state->c[0] = state->c[max_index];
+
+ return status;
+ }
+ }
+}
+
+
+static
+void
+cspline_free (void * vstate)
+{
+ cspline_state_t *state = (cspline_state_t *) vstate;
+
+ free (state->c);
+ free (state->g);
+ free (state->diag);
+ free (state->offdiag);
+ free (state);
+}
+
+/* function for common coefficient determination
+ */
+static inline void
+coeff_calc (const double c_array[], double dy, double dx, size_t index,
+ double * b, double * c, double * d)
+{
+ const double c_i = c_array[index];
+ const double c_ip1 = c_array[index + 1];
+ *b = (dy / dx) - dx * (c_ip1 + 2.0 * c_i) / 3.0;
+ *c = c_i;
+ *d = (c_ip1 - c_i) / (3.0 * dx);
+}
+
+
+static
+int
+cspline_eval (const void * vstate,
+ const double x_array[], const double y_array[], size_t size,
+ double x,
+ gsl_interp_accel * a,
+ double *y)
+{
+ const cspline_state_t *state = (const cspline_state_t *) vstate;
+
+ double x_lo, x_hi;
+ double dx;
+ size_t index;
+
+ if (a != 0)
+ {
+ index = gsl_interp_accel_find (a, x_array, size, x);
+ }
+ else
+ {
+ index = gsl_interp_bsearch (x_array, x, 0, size - 1);
+ }
+
+ /* evaluate */
+ x_hi = x_array[index + 1];
+ x_lo = x_array[index];
+ dx = x_hi - x_lo;
+ if (dx > 0.0)
+ {
+ const double y_lo = y_array[index];
+ const double y_hi = y_array[index + 1];
+ const double dy = y_hi - y_lo;
+ double delx = x - x_lo;
+ double b_i, c_i, d_i;
+ coeff_calc(state->c, dy, dx, index, &b_i, &c_i, &d_i);
+ *y = y_lo + delx * (b_i + delx * (c_i + delx * d_i));
+ return GSL_SUCCESS;
+ }
+ else
+ {
+ *y = 0.0;
+ return GSL_EINVAL;
+ }
+}
+
+
+static
+int
+cspline_eval_deriv (const void * vstate,
+ const double x_array[], const double y_array[], size_t size,
+ double x,
+ gsl_interp_accel * a,
+ double *dydx)
+{
+ const cspline_state_t *state = (const cspline_state_t *) vstate;
+
+ double x_lo, x_hi;
+ double dx;
+ size_t index;
+
+ if (a != 0)
+ {
+ index = gsl_interp_accel_find (a, x_array, size, x);
+ }
+ else
+ {
+ index = gsl_interp_bsearch (x_array, x, 0, size - 1);
+ }
+
+ /* evaluate */
+ x_hi = x_array[index + 1];
+ x_lo = x_array[index];
+ dx = x_hi - x_lo;
+ if (dx > 0.0)
+ {
+ const double y_lo = y_array[index];
+ const double y_hi = y_array[index + 1];
+ const double dy = y_hi - y_lo;
+ double delx = x - x_lo;
+ double b_i, c_i, d_i;
+ coeff_calc(state->c, dy, dx, index, &b_i, &c_i, &d_i);
+ *dydx = b_i + delx * (2.0 * c_i + 3.0 * d_i * delx);
+ return GSL_SUCCESS;
+ }
+ else
+ {
+ *dydx = 0.0;
+ return GSL_FAILURE;
+ }
+}
+
+
+static
+int
+cspline_eval_deriv2 (const void * vstate,
+ const double x_array[], const double y_array[], size_t size,
+ double x,
+ gsl_interp_accel * a,
+ double * y_pp)
+{
+ const cspline_state_t *state = (const cspline_state_t *) vstate;
+
+ double x_lo, x_hi;
+ double dx;
+ size_t index;
+
+ if (a != 0)
+ {
+ index = gsl_interp_accel_find (a, x_array, size, x);
+ }
+ else
+ {
+ index = gsl_interp_bsearch (x_array, x, 0, size - 1);
+ }
+
+ /* evaluate */
+ x_hi = x_array[index + 1];
+ x_lo = x_array[index];
+ dx = x_hi - x_lo;
+ if (dx > 0.0)
+ {
+ const double y_lo = y_array[index];
+ const double y_hi = y_array[index + 1];
+ const double dy = y_hi - y_lo;
+ double delx = x - x_lo;
+ double b_i, c_i, d_i;
+ coeff_calc(state->c, dy, dx, index, &b_i, &c_i, &d_i);
+ *y_pp = 2.0 * c_i + 6.0 * d_i * delx;
+ return GSL_SUCCESS;
+ }
+ else
+ {
+ *y_pp = 0.0;
+ return GSL_FAILURE;
+ }
+}
+
+
+static
+int
+cspline_eval_integ (const void * vstate,
+ const double x_array[], const double y_array[], size_t size,
+ gsl_interp_accel * acc,
+ double a, double b,
+ double * result)
+{
+ const cspline_state_t *state = (const cspline_state_t *) vstate;
+
+ size_t i, index_a, index_b;
+
+ if (acc != 0)
+ {
+ index_a = gsl_interp_accel_find (acc, x_array, size, a);
+ index_b = gsl_interp_accel_find (acc, x_array, size, b);
+ }
+ else
+ {
+ index_a = gsl_interp_bsearch (x_array, a, 0, size - 1);
+ index_b = gsl_interp_bsearch (x_array, b, 0, size - 1);
+ }
+
+ *result = 0.0;
+
+ /* interior intervals */
+ for(i=index_a; i<=index_b; i++) {
+ const double x_hi = x_array[i + 1];
+ const double x_lo = x_array[i];
+ const double y_lo = y_array[i];
+ const double y_hi = y_array[i + 1];
+ const double dx = x_hi - x_lo;
+ const double dy = y_hi - y_lo;
+ if(dx != 0.0) {
+ double b_i, c_i, d_i;
+ coeff_calc(state->c, dy, dx, i, &b_i, &c_i, &d_i);
+
+ if (i == index_a || i == index_b)
+ {
+ double x1 = (i == index_a) ? a : x_lo;
+ double x2 = (i == index_b) ? b : x_hi;
+ *result += integ_eval(y_lo, b_i, c_i, d_i, x_lo, x1, x2);
+ }
+ else
+ {
+ *result += dx * (y_lo + dx*(0.5*b_i + dx*(c_i/3.0 + 0.25*d_i*dx)));
+ }
+ }
+ else {
+ *result = 0.0;
+ return GSL_FAILURE;
+ }
+ }
+
+ return GSL_SUCCESS;
+}
+
+static const gsl_interp_type cspline_type =
+{
+ "cspline",
+ 3,
+ &cspline_alloc,
+ &cspline_init,
+ &cspline_eval,
+ &cspline_eval_deriv,
+ &cspline_eval_deriv2,
+ &cspline_eval_integ,
+ &cspline_free
+};
+
+const gsl_interp_type * gsl_interp_cspline = &cspline_type;
+
+static const gsl_interp_type cspline_periodic_type =
+{
+ "cspline-periodic",
+ 2,
+ &cspline_alloc,
+ &cspline_init_periodic,
+ &cspline_eval,
+ &cspline_eval_deriv,
+ &cspline_eval_deriv2,
+ &cspline_eval_integ,
+ &cspline_free
+};
+
+const gsl_interp_type * gsl_interp_cspline_periodic = &cspline_periodic_type;
+
+
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