lmcurve - Man Page

Levenberg-Marquardt least-squares fit of a curve (t,y)

Synopsis

#include <lmcurve.h>

void lmcurve( const int n_par, double *par, const int m_dat, const double *t, const double *y, double (*f)( const double ti, const double *par ), const lm_control_struct *control, lm_status_struct *status);

void lmcurve_tyd( const int n_par, double *par, const int m_dat, const double *t, const double *y, const double *dy, double (*f)( const double ti, const double *par ), const lm_control_struct *control, lm_status_struct *status);

extern const lm_control_struct lm_control_double;

extern const lm_control_struct lm_control_float;

extern const char *lm_infmsg[];

extern const char *lm_shortmsg[];

Description

lmcurve() and lmcurve_tyd() wrap the more generic minimization function lmmin(), for use in curve fitting.

lmcurve() determines a vector par that minimizes the sum of squared elements of a residue vector r[i] := y[i] - f(t[i];par). Typically, lmcurve() is used to approximate a data set t,y by a parametric function f(ti;par). On success, par represents a local minimum, not necessarily a global one; it may depend on its starting value.

lmcurve_tyd() does the same for a data set t,y,dy, where dy represents the standard deviation of empirical data y. Residues are computed as r[i] := (y[i] - f(t[i];par))/dy[i]. Users must ensure that all dy[i] are positive.

Function arguments:

n_par: Number of free variables. Length of parameter vector par.
par: Parameter vector. On input, it must contain a reasonable guess. On output, it contains the solution found to minimize ||r||.
m_dat: Number of data points. Length of vectors t and y. Must statisfy n_par <= m_dat.
t: Array of length m_dat. Contains the abcissae (time, or "x") for which function f will be evaluated.
y: Array of length m_dat. Contains the ordinate values that shall be fitted.
dy: Only in lmcurve_tyd(). Array of length m_dat. Contains the standard deviations of the values y.
f: A user-supplied parametric function f(ti;par).
control: Parameter collection for tuning the fit procedure. In most cases, the default &lm_control_double is adequate. If f is only computed with single-precision accuracy, &lm_control_float should be used. Parameters are explained in lmmin(3).
status: A record used to return information about the minimization process: For details, see lmmin(3).

Example

Fit a data set y(x) by a curve f(x;p):

    #include "lmcurve.h"
    #include <stdio.h>

    /* model function: a parabola */

    double f( double t, const double *p )
    {
        return p[0] + p[1]*t + p[2]*t*t;
    }

    int main()
    {
        int n = 3; /* number of parameters in model function f */
        double par[3] = { 100, 0, -10 }; /* really bad starting value */

        /* data points: a slightly distorted standard parabola */
        int m = 9;
        int i;
        double t[9] = { -4., -3., -2., -1.,  0., 1.,  2.,  3.,  4. };
        double y[9] = { 16.6, 9.9, 4.4, 1.1, 0., 1.1, 4.2, 9.3, 16.4 };

        lm_control_struct control = lm_control_double;
        lm_status_struct status;
        control.verbosity = 7;

        printf( "Fitting ...\n" );
        lmcurve( n, par, m, t, y, f, &control, &status );

        printf( "Results:\n" );
        printf( "status after %d function evaluations:\n  %s\n",
                status.nfev, lm_infmsg[status.outcome] );

        printf("obtained parameters:\n");
        for ( i = 0; i < n; ++i)
            printf("  par[%i] = %12g\n", i, par[i]);
        printf("obtained norm:\n  %12g\n", status.fnorm );

        printf("fitting data as follows:\n");
        for ( i = 0; i < m; ++i)
            printf( "  t[%2d]=%4g y=%6g fit=%10g residue=%12g\n",
                    i, t[i], y[i], f(t[i],par), y[i] - f(t[i],par) );

        return 0;
    }

lmcurve - Man Page

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