/* 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 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., 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_EINVAL;
}
}
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_EINVAL;
}
}
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_EINVAL;
}
}
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;