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#include <stdlib.h>
#include <stdio.h>
#include <gsl/gsl_rng.h>
#include <gsl/gsl_randist.h>
#include <gsl/gsl_matrix.h>
#include <gsl/gsl_vector.h>
#include <gsl/gsl_blas.h>
#include <gsl/gsl_multifit_nlinear.h>
/* number of data points to fit */
#define N 40
struct data {
size_t n;
double * y;
};
int
expb_f (const gsl_vector * x, void *data,
gsl_vector * f)
{
size_t n = ((struct data *)data)->n;
double *y = ((struct data *)data)->y;
double A = gsl_vector_get (x, 0);
double lambda = gsl_vector_get (x, 1);
double b = gsl_vector_get (x, 2);
size_t i;
for (i = 0; i < n; i++)
{
/* Model Yi = A * exp(-lambda * i) + b */
double t = i;
double Yi = A * exp (-lambda * t) + b;
gsl_vector_set (f, i, Yi - y[i]);
}
return GSL_SUCCESS;
}
int
expb_df (const gsl_vector * x, void *data,
gsl_matrix * J)
{
size_t n = ((struct data *)data)->n;
double A = gsl_vector_get (x, 0);
double lambda = gsl_vector_get (x, 1);
size_t i;
for (i = 0; i < n; i++)
{
/* Jacobian matrix J(i,j) = dfi / dxj, */
/* where fi = (Yi - yi)/sigma[i], */
/* Yi = A * exp(-lambda * i) + b */
/* and the xj are the parameters (A,lambda,b) */
double t = i;
double e = exp(-lambda * t);
gsl_matrix_set (J, i, 0, e);
gsl_matrix_set (J, i, 1, -t * A * e);
gsl_matrix_set (J, i, 2, 1.0);
}
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 rcond;
/* compute reciprocal condition number of J(x) */
gsl_multifit_nlinear_rcond(&rcond, w);
fprintf(stderr, "iter %2zu: A = %.4f, lambda = %.4f, b = %.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),
1.0 / rcond,
gsl_blas_dnrm2(f));
}
int
main (void)
{
const gsl_multifit_nlinear_type *T = gsl_multifit_nlinear_trust;
gsl_multifit_nlinear_workspace *w;
gsl_multifit_nlinear_fdf fdf;
gsl_multifit_nlinear_parameters fdf_params =
gsl_multifit_nlinear_default_parameters();
const size_t n = N;
const size_t p = 3;
gsl_vector *f;
gsl_matrix *J;
gsl_matrix *covar = gsl_matrix_alloc (p, p);
double y[N], weights[N];
struct data d = { n, y };
double x_init[3] = { 1.0, 1.0, 0.0 }; /* starting values */
gsl_vector_view x = gsl_vector_view_array (x_init, p);
gsl_vector_view wts = gsl_vector_view_array(weights, n);
gsl_rng * r;
double chisq, chisq0;
int status, info;
size_t i;
const double xtol = 1e-8;
const double gtol = 1e-8;
const double ftol = 0.0;
gsl_rng_env_setup();
r = gsl_rng_alloc(gsl_rng_default);
/* define the function to be minimized */
fdf.f = expb_f;
fdf.df = expb_df; /* set to NULL for finite-difference Jacobian */
fdf.fvv = NULL; /* not using geodesic acceleration */
fdf.n = n;
fdf.p = p;
fdf.params = &d;
/* this is the data to be fitted */
for (i = 0; i < n; i++)
{
double t = i;
double yi = 1.0 + 5 * exp (-0.1 * t);
double si = 0.1 * yi;
double dy = gsl_ran_gaussian(r, si);
weights[i] = 1.0 / (si * si);
y[i] = yi + dy;
printf ("data: %zu %g %g\n", i, y[i], si);
};
/* allocate workspace with default parameters */
w = gsl_multifit_nlinear_alloc (T, &fdf_params, n, p);
/* initialize solver with starting point and weights */
gsl_multifit_nlinear_winit (&x.vector, &wts.vector, &fdf, w);
/* compute initial cost function */
f = gsl_multifit_nlinear_residual(w);
gsl_blas_ddot(f, f, &chisq0);
/* solve the system with a maximum of 20 iterations */
status = gsl_multifit_nlinear_driver(20, xtol, gtol, ftol,
callback, NULL, &info, w);
/* compute covariance of best fit parameters */
J = gsl_multifit_nlinear_jac(w);
gsl_multifit_nlinear_covar (J, 0.0, covar);
/* compute final cost */
gsl_blas_ddot(f, f, &chisq);
#define FIT(i) gsl_vector_get(w->x, i)
#define ERR(i) sqrt(gsl_matrix_get(covar,i,i))
fprintf(stderr, "summary from method '%s/%s'\n",
gsl_multifit_nlinear_name(w),
gsl_multifit_nlinear_trs_name(w));
fprintf(stderr, "number of iterations: %zu\n",
gsl_multifit_nlinear_niter(w));
fprintf(stderr, "function evaluations: %zu\n", fdf.nevalf);
fprintf(stderr, "Jacobian evaluations: %zu\n", fdf.nevaldf);
fprintf(stderr, "reason for stopping: %s\n",
(info == 1) ? "small step size" : "small gradient");
fprintf(stderr, "initial |f(x)| = %f\n", sqrt(chisq0));
fprintf(stderr, "final |f(x)| = %f\n", sqrt(chisq));
{
double dof = n - p;
double c = GSL_MAX_DBL(1, sqrt(chisq / dof));
fprintf(stderr, "chisq/dof = %g\n", chisq / dof);
fprintf (stderr, "A = %.5f +/- %.5f\n", FIT(0), c*ERR(0));
fprintf (stderr, "lambda = %.5f +/- %.5f\n", FIT(1), c*ERR(1));
fprintf (stderr, "b = %.5f +/- %.5f\n", FIT(2), c*ERR(2));
}
fprintf (stderr, "status = %s\n", gsl_strerror (status));
gsl_multifit_nlinear_free (w);
gsl_matrix_free (covar);
gsl_rng_free (r);
return 0;
}
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