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<title>GNU Scientific Library – Reference Manual: Example programs for B-splines</title>
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<a name="Example-programs-for-B_002dsplines"></a>
<div class="header">
<p>
Next: <a href="B_002dSpline-References-and-Further-Reading.html#B_002dSpline-References-and-Further-Reading" accesskey="n" rel="next">B-Spline References and Further Reading</a>, Previous: <a href="Working-with-the-Greville-abscissae.html#Working-with-the-Greville-abscissae" accesskey="p" rel="previous">Working with the Greville abscissae</a>, Up: <a href="Basis-Splines.html#Basis-Splines" accesskey="u" rel="up">Basis Splines</a> [<a href="Function-Index.html#Function-Index" title="Index" rel="index">Index</a>]</p>
</div>
<hr>
<a name="Examples-30"></a>
<h3 class="section">40.7 Examples</h3>
<a name="index-basis-splines_002c-examples"></a>
<p>The following program computes a linear least squares fit to data using
cubic B-spline basis functions with uniform breakpoints. The data is
generated from the curve <em>y(x) = \cos{(x)} \exp{(-x/10)}</em> on
the interval <em>[0, 15]</em> with Gaussian noise added.
</p>
<div class="example">
<pre class="verbatim">#include <stdio.h>
#include <stdlib.h>
#include <math.h>
#include <gsl/gsl_bspline.h>
#include <gsl/gsl_multifit.h>
#include <gsl/gsl_rng.h>
#include <gsl/gsl_randist.h>
#include <gsl/gsl_statistics.h>
/* number of data points to fit */
#define N 200
/* number of fit coefficients */
#define NCOEFFS 12
/* nbreak = ncoeffs + 2 - k = ncoeffs - 2 since k = 4 */
#define NBREAK (NCOEFFS - 2)
int
main (void)
{
const size_t n = N;
const size_t ncoeffs = NCOEFFS;
const size_t nbreak = NBREAK;
size_t i, j;
gsl_bspline_workspace *bw;
gsl_vector *B;
double dy;
gsl_rng *r;
gsl_vector *c, *w;
gsl_vector *x, *y;
gsl_matrix *X, *cov;
gsl_multifit_linear_workspace *mw;
double chisq, Rsq, dof, tss;
gsl_rng_env_setup();
r = gsl_rng_alloc(gsl_rng_default);
/* allocate a cubic bspline workspace (k = 4) */
bw = gsl_bspline_alloc(4, nbreak);
B = gsl_vector_alloc(ncoeffs);
x = gsl_vector_alloc(n);
y = gsl_vector_alloc(n);
X = gsl_matrix_alloc(n, ncoeffs);
c = gsl_vector_alloc(ncoeffs);
w = gsl_vector_alloc(n);
cov = gsl_matrix_alloc(ncoeffs, ncoeffs);
mw = gsl_multifit_linear_alloc(n, ncoeffs);
printf("#m=0,S=0\n");
/* this is the data to be fitted */
for (i = 0; i < n; ++i)
{
double sigma;
double xi = (15.0 / (N - 1)) * i;
double yi = cos(xi) * exp(-0.1 * xi);
sigma = 0.1 * yi;
dy = gsl_ran_gaussian(r, sigma);
yi += dy;
gsl_vector_set(x, i, xi);
gsl_vector_set(y, i, yi);
gsl_vector_set(w, i, 1.0 / (sigma * sigma));
printf("%f %f\n", xi, yi);
}
/* use uniform breakpoints on [0, 15] */
gsl_bspline_knots_uniform(0.0, 15.0, bw);
/* construct the fit matrix X */
for (i = 0; i < n; ++i)
{
double xi = gsl_vector_get(x, i);
/* compute B_j(xi) for all j */
gsl_bspline_eval(xi, B, bw);
/* fill in row i of X */
for (j = 0; j < ncoeffs; ++j)
{
double Bj = gsl_vector_get(B, j);
gsl_matrix_set(X, i, j, Bj);
}
}
/* do the fit */
gsl_multifit_wlinear(X, w, y, c, cov, &chisq, mw);
dof = n - ncoeffs;
tss = gsl_stats_wtss(w->data, 1, y->data, 1, y->size);
Rsq = 1.0 - chisq / tss;
fprintf(stderr, "chisq/dof = %e, Rsq = %f\n",
chisq / dof, Rsq);
/* output the smoothed curve */
{
double xi, yi, yerr;
printf("#m=1,S=0\n");
for (xi = 0.0; xi < 15.0; xi += 0.1)
{
gsl_bspline_eval(xi, B, bw);
gsl_multifit_linear_est(B, c, cov, &yi, &yerr);
printf("%f %f\n", xi, yi);
}
}
gsl_rng_free(r);
gsl_bspline_free(bw);
gsl_vector_free(B);
gsl_vector_free(x);
gsl_vector_free(y);
gsl_matrix_free(X);
gsl_vector_free(c);
gsl_vector_free(w);
gsl_matrix_free(cov);
gsl_multifit_linear_free(mw);
return 0;
} /* main() */
</pre></div>
<p>The output can be plotted with <small>GNU</small> <code>graph</code>.
</p>
<div class="example">
<pre class="example">$ ./a.out > bspline.txt
chisq/dof = 1.118217e+00, Rsq = 0.989771
$ graph -T ps -X x -Y y -x 0 15 -y -1 1.3 < bspline.txt > bspline.ps
</pre></div>
<hr>
<div class="header">
<p>
Next: <a href="B_002dSpline-References-and-Further-Reading.html#B_002dSpline-References-and-Further-Reading" accesskey="n" rel="next">B-Spline References and Further Reading</a>, Previous: <a href="Working-with-the-Greville-abscissae.html#Working-with-the-Greville-abscissae" accesskey="p" rel="previous">Working with the Greville abscissae</a>, Up: <a href="Basis-Splines.html#Basis-Splines" accesskey="u" rel="up">Basis Splines</a> [<a href="Function-Index.html#Function-Index" title="Index" rel="index">Index</a>]</p>
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