NAME

lmcurve2 - Levenberg-Marquardt least-squares fit of a curve (t,y,dy)

SYNOPSIS

#include <lmcurve2.h>

void lmcurve2(
const int n_par, double *par,
double *parerr, double *covar,
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

lmcurve2() wraps the more generic minimization function lmmin2(), for use in curve fitting.

lmcurve2() determines a vector par that minimizes the sum of squared elements of a residue vector r[i] := (y[i] - f(t[i];par)) / dy[i]. Typically, lmcurve2() is used to approximate a data set t,y,dy, where dy represents the standard deviation of empirical data 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. 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||.
parerr: Parameter uncertainties vector. Array of length n_par or NULL. On output, unless it or covar is NULL, it contains the weighted parameter uncertainties for the found parameters.
covar: Covariance matrix. Array of length n_par * n_par or NULL. On output, unless it is NULL, it contains the covariance matrix.
m_dat: Number of data points. Length of vectors t, y, dy. 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: 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 lmmin2(3).
status: A record used to return information about the minimization process: For details, see lmmin2(3).

EXAMPLE

Fit a data set y(x) with standard deviations dy(x) by a curve f(x;p):

    #include "lmcurve2.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 */
        double parerr[3];
        double covar[3*3];
        /* 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 };
        double dy[9] = { 4, 3, 2, 1, 2, 3, 4, 5, 6 };
        lm_control_struct control = lm_control_double;
        lm_status_struct status;
        control.verbosity = 1;
        printf( "Fitting ...\n" );
        /* now the call to lmfit */
        lmcurve2( n, par, parerr, covar, m, t, y, dy, 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  uncertainty = %12g\n", i, par[i], parerr[i]);
        printf("obtained norm:\n  %12g\n", status.fnorm );
        printf("fitting data as follows:\n");
        for ( i = 0; i < m; ++i)
            printf(
            "  t[%1d]=%2g y=%5.1f+-%4.1f fit=%8.5f residue=%8.4f weighed=%8.4f\n",
            i, t[i], y[i], dy[i], f(t[i],par), y[i] - f(t[i],par),
            (y[i] - f(t[i],par))/dy[i] );
        return 0;
    }

COPYING

Software: FreeBSD License

Documentation: Creative Commons Attribution Share Alike

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