lmmin - Levenberg-Marquardt least-squares minimization
#include <lmmin.h>
void lmmin( const int n_par, double *par, const int m_dat, const void *data, void *evaluate( const double *par, const int m_dat, const void *data, double *fvec, int *userbreak), 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[];
lmmin() determines a vector par that minimizes the sum of squared elements of a vector fvec that is computed by a user-supplied function evaluate(). On success, par represents a local minimum, not necessarily a global one; it may depend on its starting value.
For applications in curve fitting, the wrapper function lmcurve(3) offers a simplified API.
The Levenberg-Marquardt minimization starts with a steepest-descent exploration of the parameter space, and achieves rapid convergence by crossing over into the Newton-Gauss method.
Function arguments:
Number of free variables. Length of parameter vector par.
Parameter vector. On input, it must contain a reasonable guess. On output, it contains the solution found to minimize ||fvec||.
Length of vector fvec. Must statisfy n_par <= m_dat.
This pointer is ignored by the fit algorithm, except for appearing as an argument in all calls to the user-supplied routine evaluate.
Pointer to a user-supplied function that computes m_dat elements of vector fvec for a given parameter vector par. If evaluate return with *userbreak set to a negative value, lmmin() will interrupt the fitting and terminate.
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. See also below, NOTES on initializing parameter records.
control has the following members (for more details, see the source file lmstruct.h):
Relative error desired in the sum of squares. Recommended setting: somewhat above machine precision; less if fvec is computed with reduced accuracy.
Relative error between last two approximations. Recommended setting: as ftol.
A measure for degeneracy. Recommended setting: as ftol.
Step used to calculate the Jacobian. Recommended setting: as ftol, but definitely less than the accuracy of fvec.
Initial bound to steps in the outer loop, generally between 0.01 and 100; recommended value is 100.
Used to set the maximum number of function evaluations to patience*n_par.
Logical switch (0 or 1). If 1, then scale parameters to their initial value. This is the recommended setting.
Progress messages will be written to this file. Typically stdout or stderr. The value NULL will be interpreted as stdout.
If nonzero, some progress information from within the LM algorithm is written to control.stream.
-1, or maximum number of parameters to print.
-1, or maximum number of residuals to print.
A record used to return information about the minimization process:
Norm of the vector fvec;
Actual number of iterations;
Status of minimization; for the corresponding text message, print lm_infmsg[status.outcome]; for a short code, print lm_shortmsg[status.outcome].
Set when termination has been forced by the user-supplied routine evaluate.
The parameter record control should always be initialized from supplied default records:
lm_control_struct control = lm_control_double; /* or _float */After this, parameters may be overwritten:
control.patience = 500; /* allow more iterations */
control.verbosity = 15; /* for verbose monitoring */An application written this way is guaranteed to work even if new parameters are added to lm_control_struct.
Conversely, addition of parameters is not considered an API change; it may happen without increment of the major version number.
Fit a data set y(t) by a function f(t;p) where t is a two-dimensional vector:
#include "lmmin.h"
#include <stdio.h>
/* fit model: a plane p0 + p1*tx + p2*tz */
double f( double tx, double tz, const double *p )
{
return p[0] + p[1]*tx + p[2]*tz;
}
/* data structure to transmit data arays and fit model */
typedef struct {
double *tx, *tz;
double *y;
double (*f)( double tx, double tz, const double *p );
} data_struct;
/* function evaluation, determination of residues */
void evaluate_surface( const double *par, int m_dat,
const void *data, double *fvec, int *userbreak )
{
/* for readability, explicit type conversion */
data_struct *D;
D = (data_struct*)data;
int i;
for ( i = 0; i < m_dat; i++ )
fvec[i] = D->y[i] - D->f( D->tx[i], D->tz[i], par );
}
int main()
{
/* parameter vector */
int n_par = 3; /* number of parameters in model function f */
double par[3] = { -1, 0, 1 }; /* arbitrary starting value */
/* data points */
int m_dat = 4;
double tx[4] = { -1, -1, 1, 1 };
double tz[4] = { -1, 1, -1, 1 };
double y[4] = { 0, 1, 1, 2 };
data_struct data = { tx, tz, y, f };
/* auxiliary parameters */
lm_status_struct status;
lm_control_struct control = lm_control_double;
control.verbosity = 3;
/* perform the fit */
printf( "Fitting:\n" );
lmmin( n_par, par, m_dat, (const void*) &data, evaluate_surface,
&control, &status );
/* print results */
printf( "\nResults:\n" );
printf( "status after %d function evaluations:\n %s\n",
status.nfev, lm_infmsg[status.outcome] );
printf("obtained parameters:\n");
int i;
for ( i=0; i<n_par; ++i )
printf(" par[%i] = %12g\n", i, par[i]);
printf("obtained norm:\n %12g\n", status.fnorm );
printf("fitting data as follows:\n");
double ff;
for ( i=0; i<m_dat; ++i ){
ff = f(tx[i], tz[i], par);
printf( " t[%2d]=%12g,%12g y=%12g fit=%12g residue=%12g\n",
i, tx[i], tz[i], y[i], ff, y[i] - ff );
}
return 0;
}For more examples, see the homepage and directories demo/ and test/ in the source distribution.
Copyright (C): 1980-1999 University of Chicago 2004-2015 Joachim Wuttke, Forschungszentrum Juelich GmbH
Software: FreeBSD License
Documentation: Creative Commons Attribution Share Alike
Homepage: http://apps.jcns.fz-juelich.de/lmfit
Please send bug reports and suggestions to the author <j.wuttke@fz-juelich.de>.