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Title "lmmin 3" For nroff, turn off justification. Always turn off hyphenation; it makes
way too many mistakes in technical documents.
.nh lmmin - Levenberg-Marquardt least-squares minimization Header "SYNOPSIS" \fB#include <lmmin.h>

\fBvoid 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 );

\fBextern const lm_control_struct lm_control_double;

\fBextern const lm_control_struct lm_control_float;

\fBextern const char *lm_infmsg[];

\fBextern const char *lm_shortmsg[];

Header "DESCRIPTION" \fB\f(BIlmmin() 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 \f(BIlmcurve\|(3) offers a simplified \s-1API.\s0

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:

Item "n_par" Number of free variables. Length of parameter vector par. Item "par" Parameter vector. On input, it must contain a reasonable guess. On output, it contains the solution found to minimize ||fvec||. Item "m_dat" Length of vector fvec. Must statisfy n_par <= m_dat. Item "data" This pointer is ignored by the fit algorithm, except for appearing as an argument in all calls to the user-supplied routine evaluate. Item "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, \f(BIlmmin() will interrupt the fitting and terminate.

"control" 4

Item "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, \fI&lm_control_float should be used. See also below, \s-1NOTES\s0 on initializing parameter records. .Sp \fIcontrol has the following members (for more details, see the source file lmstruct.h): Item "double control.ftol" Relative error desired in the sum of squares. Recommended setting: somewhat above machine precision; less if fvec is computed with reduced accuracy. Item "double control.xtol" Relative error between last two approximations. Recommended setting: as ftol. Item "double control.gtol" A measure for degeneracy. Recommended setting: as ftol. Item "double control.epsilon" Step used to calculate the Jacobian. Recommended setting: as ftol, but definitely less than the accuracy of fvec. Item "double control.stepbound" Initial bound to steps in the outer loop, generally between 0.01 and 100; recommended value is 100. Item "int control.patience" Used to set the maximum number of function evaluations to patience*n_par. Item "int control.scale_diag" Logical switch (0 or 1). If 1, then scale parameters to their initial value. This is the recommended setting. Item "FILE* control.msgfile" Progress messages will be written to this file. Typically stdout or stderr. The value \s-1NULL\s0 will be interpreted as stdout. Item "int control.verbosity" If nonzero, some progress information from within the \s-1LM\s0 algorithm is written to control.stream. Item "int control.n_maxpri" \-1, or maximum number of parameters to print. Item "int control.m_maxpri" \-1, or maximum number of residuals to print. Item "status" A record used to return information about the minimization process: Item "double status.fnorm" Norm of the vector fvec; Item "int status.nfev" Actual number of iterations; Item "int status.outcome" Status of minimization; for the corresponding text message, print lm_infmsg[status.outcome]; for a short code, print lm_shortmsg[status.outcome]. Item "int status.userbreak" Set when termination has been forced by the user-supplied routine evaluate. Header "NOTES" Subsection "Initializing parameter records." The parameter record control should always be initialized from supplied default records:

.Vb 1 lm_control_struct control = lm_control_double; /* or _float */ .Ve

After this, parameters may be overwritten:

.Vb 2 control.patience = 500; /* allow more iterations */ control.verbosity = 15; /* for verbose monitoring */ .Ve

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 \s-1API\s0 change; it may happen without increment of the major version number.

Header "EXAMPLES" Subsection "Fitting a surface" Fit a data set y(t) by a function f(t;p) where t is a two-dimensional vector:

.Vb 2 #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:\en" ); lmmin( n_par, par, m_dat, (const void*) &data, evaluate_surface, &control, &status ); \& /* print results */ printf( "\enResults:\en" ); printf( "status after %d function evaluations:\en %s\en", status.nfev, lm_infmsg[status.outcome] ); \& printf("obtained parameters:\en"); int i; for ( i=0; i<n_par; ++i ) printf(" par[%i] = %12g\en", i, par[i]); printf("obtained norm:\en %12g\en", status.fnorm ); \& printf("fitting data as follows:\en"); 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\en", i, tx[i], tz[i], y[i], ff, y[i] - ff ); } \& return 0; } .Ve

Subsection "More examples" For more examples, see the homepage and directories demo/ and test/ in the source distribution. Header "COPYING" Copyright (C): 1980-1999 University of Chicago 2004-2015 Joachim Wuttke, Forschungszentrum Juelich GmbH

Software: FreeBSD License

Documentation: Creative Commons Attribution Share Alike

Header "SEE ALSO" lmcurve(3)

Homepage: http://apps.jcns.fz-juelich.de/lmfit

Header "BUGS" Please send bug reports and suggestions to the author <j.wuttke@fz-juelich.de>.