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#include <stdlib.h>
#include <stdio.h>
#include <gsl/gsl_vector.h>
#include <gsl/gsl_matrix.h>
#include <gsl/gsl_blas.h>
#include <gsl/gsl_multifit_nlinear.h>
#include <gsl/gsl_rng.h>
#include <gsl/gsl_randist.h>
struct data
{
double *t;
double *y;
size_t n;
};
/* model function: a * exp( -1/2 * [ (t - b) / c ]^2 ) */
double
gaussian(const double a, const double b, const double c, const double t)
{
const double z = (t - b) / c;
return (a * exp(-0.5 * z * z));
}
int
func_f (const gsl_vector * x, void *params, gsl_vector * f)
{
struct data *d = (struct data *) params;
double a = gsl_vector_get(x, 0);
double b = gsl_vector_get(x, 1);
double c = gsl_vector_get(x, 2);
size_t i;
for (i = 0; i < d->n; ++i)
{
double ti = d->t[i];
double yi = d->y[i];
double y = gaussian(a, b, c, ti);
gsl_vector_set(f, i, yi - y);
}
return GSL_SUCCESS;
}
int
func_df (const gsl_vector * x, void *params, gsl_matrix * J)
{
struct data *d = (struct data *) params;
double a = gsl_vector_get(x, 0);
double b = gsl_vector_get(x, 1);
double c = gsl_vector_get(x, 2);
size_t i;
for (i = 0; i < d->n; ++i)
{
double ti = d->t[i];
double zi = (ti - b) / c;
double ei = exp(-0.5 * zi * zi);
gsl_matrix_set(J, i, 0, -ei);
gsl_matrix_set(J, i, 1, -(a / c) * ei * zi);
gsl_matrix_set(J, i, 2, -(a / c) * ei * zi * zi);
}
return GSL_SUCCESS;
}
int
func_fvv (const gsl_vector * x, const gsl_vector * v,
void *params, gsl_vector * fvv)
{
struct data *d = (struct data *) params;
double a = gsl_vector_get(x, 0);
double b = gsl_vector_get(x, 1);
double c = gsl_vector_get(x, 2);
double va = gsl_vector_get(v, 0);
double vb = gsl_vector_get(v, 1);
double vc = gsl_vector_get(v, 2);
size_t i;
for (i = 0; i < d->n; ++i)
{
double ti = d->t[i];
double zi = (ti - b) / c;
double ei = exp(-0.5 * zi * zi);
double Dab = -zi * ei / c;
double Dac = -zi * zi * ei / c;
double Dbb = a * ei / (c * c) * (1.0 - zi*zi);
double Dbc = a * zi * ei / (c * c) * (2.0 - zi*zi);
double Dcc = a * zi * zi * ei / (c * c) * (3.0 - zi*zi);
double sum;
sum = 2.0 * va * vb * Dab +
2.0 * va * vc * Dac +
vb * vb * Dbb +
2.0 * vb * vc * Dbc +
vc * vc * Dcc;
gsl_vector_set(fvv, i, sum);
}
return GSL_SUCCESS;
}
void
callback(const size_t iter, void *params,
const gsl_multifit_nlinear_workspace *w)
{
gsl_vector *f = gsl_multifit_nlinear_residual(w);
gsl_vector *x = gsl_multifit_nlinear_position(w);
double avratio = gsl_multifit_nlinear_avratio(w);
double rcond;
(void) params; /* not used */
/* compute reciprocal condition number of J(x) */
gsl_multifit_nlinear_rcond(&rcond, w);
fprintf(stderr, "iter %2zu: a = %.4f, b = %.4f, c = %.4f, |a|/|v| = %.4f cond(J) = %8.4f, |f(x)| = %.4f\n",
iter,
gsl_vector_get(x, 0),
gsl_vector_get(x, 1),
gsl_vector_get(x, 2),
avratio,
1.0 / rcond,
gsl_blas_dnrm2(f));
}
void
solve_system(gsl_vector *x, gsl_multifit_nlinear_fdf *fdf,
gsl_multifit_nlinear_parameters *params)
{
const gsl_multifit_nlinear_type *T = gsl_multifit_nlinear_trust;
const size_t max_iter = 200;
const double xtol = 1.0e-8;
const double gtol = 1.0e-8;
const double ftol = 1.0e-8;
const size_t n = fdf->n;
const size_t p = fdf->p;
gsl_multifit_nlinear_workspace *work =
gsl_multifit_nlinear_alloc(T, params, n, p);
gsl_vector * f = gsl_multifit_nlinear_residual(work);
gsl_vector * y = gsl_multifit_nlinear_position(work);
int info;
double chisq0, chisq, rcond;
/* initialize solver */
gsl_multifit_nlinear_init(x, fdf, work);
/* store initial cost */
gsl_blas_ddot(f, f, &chisq0);
/* iterate until convergence */
gsl_multifit_nlinear_driver(max_iter, xtol, gtol, ftol,
callback, NULL, &info, work);
/* store final cost */
gsl_blas_ddot(f, f, &chisq);
/* store cond(J(x)) */
gsl_multifit_nlinear_rcond(&rcond, work);
gsl_vector_memcpy(x, y);
/* print summary */
fprintf(stderr, "NITER = %zu\n", gsl_multifit_nlinear_niter(work));
fprintf(stderr, "NFEV = %zu\n", fdf->nevalf);
fprintf(stderr, "NJEV = %zu\n", fdf->nevaldf);
fprintf(stderr, "NAEV = %zu\n", fdf->nevalfvv);
fprintf(stderr, "initial cost = %.12e\n", chisq0);
fprintf(stderr, "final cost = %.12e\n", chisq);
fprintf(stderr, "final x = (%.12e, %.12e, %12e)\n",
gsl_vector_get(x, 0), gsl_vector_get(x, 1), gsl_vector_get(x, 2));
fprintf(stderr, "final cond(J) = %.12e\n", 1.0 / rcond);
gsl_multifit_nlinear_free(work);
}
int
main (void)
{
const size_t n = 300; /* number of data points to fit */
const size_t p = 3; /* number of model parameters */
const double a = 5.0; /* amplitude */
const double b = 0.4; /* center */
const double c = 0.15; /* width */
const gsl_rng_type * T = gsl_rng_default;
gsl_vector *f = gsl_vector_alloc(n);
gsl_vector *x = gsl_vector_alloc(p);
gsl_multifit_nlinear_fdf fdf;
gsl_multifit_nlinear_parameters fdf_params =
gsl_multifit_nlinear_default_parameters();
struct data fit_data;
gsl_rng * r;
size_t i;
gsl_rng_env_setup ();
r = gsl_rng_alloc (T);
fit_data.t = malloc(n * sizeof(double));
fit_data.y = malloc(n * sizeof(double));
fit_data.n = n;
/* generate synthetic data with noise */
for (i = 0; i < n; ++i)
{
double t = (double)i / (double) n;
double y0 = gaussian(a, b, c, t);
double dy = gsl_ran_gaussian (r, 0.1 * y0);
fit_data.t[i] = t;
fit_data.y[i] = y0 + dy;
}
/* define function to be minimized */
fdf.f = func_f;
fdf.df = func_df;
fdf.fvv = func_fvv;
fdf.n = n;
fdf.p = p;
fdf.params = &fit_data;
/* starting point */
gsl_vector_set(x, 0, 1.0);
gsl_vector_set(x, 1, 0.0);
gsl_vector_set(x, 2, 1.0);
fdf_params.trs = gsl_multifit_nlinear_trs_lmaccel;
solve_system(x, &fdf, &fdf_params);
/* print data and model */
{
double A = gsl_vector_get(x, 0);
double B = gsl_vector_get(x, 1);
double C = gsl_vector_get(x, 2);
for (i = 0; i < n; ++i)
{
double ti = fit_data.t[i];
double yi = fit_data.y[i];
double fi = gaussian(A, B, C, ti);
printf("%f %f %f\n", ti, yi, fi);
}
}
gsl_vector_free(f);
gsl_vector_free(x);
gsl_rng_free(r);
return 0;
}
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