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/* interpolation/steffen.c
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*
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* Copyright (C) 2014 Jean-François Caron
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*
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* This program is free software; you can redistribute it and/or modify
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* it under the terms of the GNU General Public License as published by
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* the Free Software Foundation; either version 3 of the License, or (at
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* your option) any later version.
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*
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* This program is distributed in the hope that it will be useful, but
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* WITHOUT ANY WARRANTY; without even the implied warranty of
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* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
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* General Public License for more details.
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*
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* You should have received a copy of the GNU General Public License
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* along with this program; if not, write to the Free Software
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* Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301, USA.
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*/
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/* Author: J.-F. Caron
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*
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* This interpolation method is taken from
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* M.Steffen, "A simple method for monotonic interpolation in one dimension",
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* Astron. Astrophys. 239, 443-450 (1990).
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*
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* This interpolation method guarantees monotonic interpolation functions between
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* the given data points. A consequence of this is that extremal values can only
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* occur at the data points. The interpolating function and its first derivative
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* are guaranteed to be continuous, but the second derivative is not.
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*
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* The implementation is modelled on the existing Akima interpolation method
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* previously included in GSL by Gerard Jungman.
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*/
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#include <config.h>
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#include <stdlib.h>
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#include <math.h>
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#include <gsl/gsl_math.h>
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#include <gsl/gsl_errno.h>
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#include "integ_eval.h"
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#include <gsl/gsl_interp.h>
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typedef struct
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{
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double * a; /* eqs 2-5 of paper */
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double * b;
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double * c;
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double * d;
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double * y_prime; /* eq 11 of paper */
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} steffen_state_t;
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static void steffen_free (void * vstate);
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static double steffen_copysign(const double x, const double y);
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static void *
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steffen_alloc (size_t size)
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{
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steffen_state_t *state;
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state = (steffen_state_t *) calloc (1, sizeof (steffen_state_t));
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if (state == NULL)
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{
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GSL_ERROR_NULL("failed to allocate space for state", GSL_ENOMEM);
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}
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state->a = (double *) malloc (size * sizeof (double));
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if (state->a == NULL)
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{
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steffen_free(state);
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GSL_ERROR_NULL("failed to allocate space for a", GSL_ENOMEM);
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}
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state->b = (double *) malloc (size * sizeof (double));
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if (state->b == NULL)
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{
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steffen_free(state);
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GSL_ERROR_NULL("failed to allocate space for b", GSL_ENOMEM);
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}
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state->c = (double *) malloc (size * sizeof (double));
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if (state->c == NULL)
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{
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steffen_free(state);
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GSL_ERROR_NULL("failed to allocate space for c", GSL_ENOMEM);
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}
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state->d = (double *) malloc (size * sizeof (double));
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if (state->d == NULL)
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{
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steffen_free(state);
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GSL_ERROR_NULL("failed to allocate space for d", GSL_ENOMEM);
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}
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state->y_prime = (double *) malloc (size * sizeof (double));
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if (state->y_prime == NULL)
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{
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steffen_free(state);
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GSL_ERROR_NULL("failed to allocate space for y_prime", GSL_ENOMEM);
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}
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return state;
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}
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static int
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steffen_init (void * vstate, const double x_array[],
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const double y_array[], size_t size)
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{
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steffen_state_t *state = (steffen_state_t *) vstate;
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size_t i;
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double *a = state->a;
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double *b = state->b;
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double *c = state->c;
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double *d = state->d;
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double *y_prime = state->y_prime;
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/*
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* first assign the interval and slopes for the left boundary.
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* We use the "simplest possibility" method described in the paper
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* in section 2.2
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*/
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double h0 = (x_array[1] - x_array[0]);
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double s0 = (y_array[1] - y_array[0]) / h0;
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y_prime[0] = s0;
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/* Now we calculate all the necessary s, h, p, and y' variables
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from 1 to N-2 (0 to size - 2 inclusive) */
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for (i = 1; i < (size - 1); i++)
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{
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double pi;
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/* equation 6 in the paper */
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double hi = (x_array[i+1] - x_array[i]);
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double him1 = (x_array[i] - x_array[i - 1]);
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/* equation 7 in the paper */
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double si = (y_array[i+1] - y_array[i]) / hi;
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double sim1 = (y_array[i] - y_array[i - 1]) / him1;
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/* equation 8 in the paper */
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pi = (sim1*hi + si*him1) / (him1 + hi);
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/* This is a C equivalent of the FORTRAN statement below eqn 11 */
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y_prime[i] = (steffen_copysign(1.0,sim1) + steffen_copysign(1.0,si)) *
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GSL_MIN(fabs(sim1),
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GSL_MIN(fabs(si), 0.5*fabs(pi)));
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}
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/*
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* we also need y' for the rightmost boundary; we use the
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* "simplest possibility" method described in the paper in
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* section 2.2
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*
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* y' = s_{n-1}
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*/
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y_prime[size-1] = (y_array[size - 1] - y_array[size - 2]) /
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(x_array[size - 1] - x_array[size - 2]);
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/* Now we can calculate all the coefficients for the whole range. */
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for (i = 0; i < (size - 1); i++)
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{
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double hi = (x_array[i+1] - x_array[i]);
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double si = (y_array[i+1] - y_array[i]) / hi;
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/* These are from equations 2-5 in the paper. */
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a[i] = (y_prime[i] + y_prime[i+1] - 2*si) / hi / hi;
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b[i] = (3*si - 2*y_prime[i] - y_prime[i+1]) / hi;
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c[i] = y_prime[i];
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d[i] = y_array[i];
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}
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return GSL_SUCCESS;
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}
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static void
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steffen_free (void * vstate)
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{
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steffen_state_t *state = (steffen_state_t *) vstate;
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RETURN_IF_NULL(state);
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if (state->a)
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free (state->a);
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if (state->b)
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free (state->b);
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if (state->c)
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free (state->c);
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if (state->d)
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free (state->d);
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if (state->y_prime)
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free (state->y_prime);
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free (state);
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}
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static int
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steffen_eval (const void * vstate,
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const double x_array[], const double y_array[], size_t size,
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double x, gsl_interp_accel * a, double *y)
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{
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const steffen_state_t *state = (const steffen_state_t *) vstate;
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size_t index;
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if (a != 0)
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{
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index = gsl_interp_accel_find (a, x_array, size, x);
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}
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else
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{
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index = gsl_interp_bsearch (x_array, x, 0, size - 1);
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}
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/* evaluate */
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{
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const double x_lo = x_array[index];
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const double delx = x - x_lo;
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const double a = state->a[index];
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const double b = state->b[index];
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const double c = state->c[index];
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const double d = state->d[index];
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/* Use Horner's scheme for efficient evaluation of polynomials. */
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/* *y = a*delx*delx*delx + b*delx*delx + c*delx + d; */
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*y = d + delx*(c + delx*(b + delx*a));
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return GSL_SUCCESS;
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}
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}
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static int
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steffen_eval_deriv (const void * vstate,
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const double x_array[], const double y_array[], size_t size,
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double x, gsl_interp_accel * a, double *dydx)
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{
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const steffen_state_t *state = (const steffen_state_t *) vstate;
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size_t index;
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/* DISCARD_POINTER(y_array); /\* prevent warning about unused parameter *\/ */
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if (a != 0)
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{
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index = gsl_interp_accel_find (a, x_array, size, x);
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}
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else
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{
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index = gsl_interp_bsearch (x_array, x, 0, size - 1);
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}
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/* evaluate */
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{
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double x_lo = x_array[index];
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double delx = x - x_lo;
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double a = state->a[index];
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double b = state->b[index];
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double c = state->c[index];
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/*double d = state->d[index];*/
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/* *dydx = 3*a*delx*delx*delx + 2*b*delx + c; */
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*dydx = c + delx*(2*b + delx*3*a);
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return GSL_SUCCESS;
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}
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}
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static int
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steffen_eval_deriv2 (const void * vstate,
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const double x_array[], const double y_array[], size_t size,
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double x, gsl_interp_accel * a, double *y_pp)
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{
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const steffen_state_t *state = (const steffen_state_t *) vstate;
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size_t index;
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/* DISCARD_POINTER(y_array); /\* prevent warning about unused parameter *\/ */
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if (a != 0)
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{
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index = gsl_interp_accel_find (a, x_array, size, x);
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}
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else
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{
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index = gsl_interp_bsearch (x_array, x, 0, size - 1);
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}
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/* evaluate */
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{
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const double x_lo = x_array[index];
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const double delx = x - x_lo;
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const double a = state->a[index];
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const double b = state->b[index];
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*y_pp = 6*a*delx + 2*b;
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return GSL_SUCCESS;
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}
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}
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static int
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steffen_eval_integ (const void * vstate,
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const double x_array[], const double y_array[], size_t size,
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gsl_interp_accel * acc, double a, double b,
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double * result)
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{
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/* a and b are the boundaries of the integration. */
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const steffen_state_t *state = (const steffen_state_t *) vstate;
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size_t i, index_a, index_b;
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/* Find the data points in the x_array that are nearest to the desired */
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/* a and b integration boundaries. */
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if (acc != 0)
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{
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index_a = gsl_interp_accel_find (acc, x_array, size, a);
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index_b = gsl_interp_accel_find (acc, x_array, size, b);
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}
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else
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{
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index_a = gsl_interp_bsearch (x_array, a, 0, size - 1);
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index_b = gsl_interp_bsearch (x_array, b, 0, size - 1);
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}
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*result = 0.0;
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/* Iterate over all the segments between data points and sum the */
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/* contributions into result. */
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for(i=index_a; i<=index_b; i++)
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{
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const double x_hi = x_array[i + 1];
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const double x_lo = x_array[i];
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const double dx = x_hi - x_lo;
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if(dx != 0.0)
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{
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/*
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* check if we are at a boundary point, so take the
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* a and b parameters instead of the data points.
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*/
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double x1 = (i == index_a) ? a-x_lo : 0.0;
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double x2 = (i == index_b) ? b-x_lo : x_hi-x_lo;
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67cb25 |
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*result += (1.0/4.0)*state->a[i]*(x2*x2*x2*x2 - x1*x1*x1*x1)
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+(1.0/3.0)*state->b[i]*(x2*x2*x2 - x1*x1*x1)
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+(1.0/2.0)*state->c[i]*(x2*x2 - x1*x1)
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+state->d[i]*(x2-x1);
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}
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else /* if the interval was zero, i.e. consecutive x values in data. */
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{
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*result = 0.0;
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return GSL_EINVAL;
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}
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}
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67cb25 |
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67cb25 |
return GSL_SUCCESS;
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67cb25 |
}
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67cb25 |
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67cb25 |
static double
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67cb25 |
steffen_copysign(const double x, const double y)
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67cb25 |
{
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if ((x < 0 && y > 0) || (x > 0 && y < 0))
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return -x;
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67cb25 |
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return x;
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}
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67cb25 |
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67cb25 |
static const gsl_interp_type steffen_type =
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67cb25 |
{
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"steffen",
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3,
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&steffen_alloc,
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67cb25 |
&steffen_init,
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&steffen_eval,
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&steffen_eval_deriv,
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67cb25 |
&steffen_eval_deriv2,
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67cb25 |
&steffen_eval_integ,
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67cb25 |
&steffen_free
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67cb25 |
};
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67cb25 |
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67cb25 |
const gsl_interp_type * gsl_interp_steffen = &steffen_type;
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