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Always turn off hyphenation; it makes .\" way too many mistakes in technical documents. .if n .ad l .nh .SH "NAME" lmmin \- Levenberg\-Marquardt least\-squares minimization .SH "SYNOPSIS" .IX Header "SYNOPSIS" \&\fB#include .PP \&\fBvoid lmmin( const int\fR \fIn_par\fR\fB, double *\fR\fIpar\fR\fB, const int\fR \fIm_dat\fR\fB, const\ void *\fR\fIdata\fR\fB, void *\fR\fIevaluate\fR\fB( const\ double *\fR\fIpar\fR\fB, const int \fR\fIm_dat\fR\fB, const\ void *\fR\fIdata\fR\fB, double *\fR\fIfvec\fR\fB, int *\fR\fIuserbreak\fR\fB), const\ lm_control_struct *\fR\fIcontrol\fR\fB, lm_status_struct *\fR\fIstatus\fR\fB );\fR .PP \&\fBextern const lm_control_struct lm_control_double;\fR .PP \&\fBextern const lm_control_struct lm_control_float;\fR .PP \&\fBextern const char *lm_infmsg[];\fR .PP \&\fBextern const char *lm_shortmsg[];\fR .SH "DESCRIPTION" .IX Header "DESCRIPTION" \&\fB\f(BIlmmin()\fB\fR determines a vector \fIpar\fR that minimizes the sum of squared elements of a vector \fIfvec\fR that is computed by a user-supplied function \fIevaluate\fR(). On success, \fIpar\fR represents a local minimum, not necessarily a global one; it may depend on its starting value. .PP For applications in curve fitting, the wrapper function \fB\f(BIlmcurve\fB\|(3)\fR offers a simplified \s-1API.\s0 .PP 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. .PP Function arguments: .IP "\fIn_par\fR" 4 .IX Item "n_par" Number of free variables. Length of parameter vector \fIpar\fR. .IP "\fIpar\fR" 4 .IX Item "par" Parameter vector. On input, it must contain a reasonable guess. On output, it contains the solution found to minimize ||\fIfvec\fR||. .IP "\fIm_dat\fR" 4 .IX Item "m_dat" Length of vector \fIfvec\fR. Must statisfy \fIn_par\fR <= \fIm_dat\fR. .IP "\fIdata\fR" 4 .IX Item "data" This pointer is ignored by the fit algorithm, except for appearing as an argument in all calls to the user-supplied routine \fIevaluate\fR. .IP "\fIevaluate\fR" 4 .IX Item "evaluate" Pointer to a user-supplied function that computes \fIm_dat\fR elements of vector \fIfvec\fR for a given parameter vector \fIpar\fR. If \fIevaluate\fR return with *\fIuserbreak\fR set to a negative value, \fB\f(BIlmmin()\fB\fR will interrupt the fitting and terminate. .IP "\fIcontrol\fR" 4 .IX Item "control" Parameter collection for tuning the fit procedure. In most cases, the default &\fIlm_control_double\fR is adequate. If \fIf\fR is only computed with single-precision accuracy, \&\fI&lm_control_float\fR should be used. See also below, \s-1NOTES\s0 on initializing parameter records. .Sp \&\fIcontrol\fR has the following members (for more details, see the source file \fIlmstruct.h\fR): .RS 4 .IP "\fBdouble\fR \fIcontrol.ftol\fR" 4 .IX Item "double control.ftol" Relative error desired in the sum of squares. Recommended setting: somewhat above machine precision; less if \fIfvec\fR is computed with reduced accuracy. .IP "\fBdouble\fR \fIcontrol.xtol\fR" 4 .IX Item "double control.xtol" Relative error between last two approximations. Recommended setting: as \fIftol\fR. .IP "\fBdouble\fR \fIcontrol.gtol\fR" 4 .IX Item "double control.gtol" A measure for degeneracy. Recommended setting: as \fIftol\fR. .IP "\fBdouble\fR \fIcontrol.epsilon\fR" 4 .IX Item "double control.epsilon" Step used to calculate the Jacobian. Recommended setting: as \fIftol\fR, but definitely less than the accuracy of \fIfvec\fR. .IP "\fBdouble\fR \fIcontrol.stepbound\fR" 4 .IX Item "double control.stepbound" Initial bound to steps in the outer loop, generally between 0.01 and 100; recommended value is 100. .IP "\fBint\fR \fIcontrol.patience\fR" 4 .IX Item "int control.patience" Used to set the maximum number of function evaluations to patience*n_par. .IP "\fBint\fR \fIcontrol.scale_diag\fR" 4 .IX Item "int control.scale_diag" Logical switch (0 or 1). If 1, then scale parameters to their initial value. This is the recommended setting. .IP "\fBFILE*\fR \fIcontrol.msgfile\fR" 4 .IX Item "FILE* control.msgfile" Progress messages will be written to this file. Typically \fIstdout\fR or \fIstderr\fR. The value \fI\s-1NULL\s0\fR will be interpreted as \fIstdout\fR. .IP "\fBint\fR \fIcontrol.verbosity\fR" 4 .IX Item "int control.verbosity" If nonzero, some progress information from within the \s-1LM\s0 algorithm is written to control.stream. .IP "\fBint\fR \fIcontrol.n_maxpri\fR" 4 .IX Item "int control.n_maxpri" \&\-1, or maximum number of parameters to print. .IP "\fBint\fR \fIcontrol.m_maxpri\fR" 4 .IX Item "int control.m_maxpri" \&\-1, or maximum number of residuals to print. .RE .RS 4 .RE .IP "\fIstatus\fR" 4 .IX Item "status" A record used to return information about the minimization process: .RS 4 .IP "\fBdouble\fR \fIstatus.fnorm\fR" 4 .IX Item "double status.fnorm" Norm of the vector \fIfvec\fR; .IP "\fBint\fR \fIstatus.nfev\fR" 4 .IX Item "int status.nfev" Actual number of iterations; .IP "\fBint\fR \fIstatus.outcome\fR" 4 .IX Item "int status.outcome" Status of minimization; for the corresponding text message, print \fIlm_infmsg\fR\fB[\fR\fIstatus.outcome\fR\fB]\fR; for a short code, print \fIlm_shortmsg\fR\fB[\fR\fIstatus.outcome\fR\fB]\fR. .IP "\fBint\fR \fIstatus.userbreak\fR" 4 .IX Item "int status.userbreak" Set when termination has been forced by the user-supplied routine \fIevaluate\fR. .RE .RS 4 .RE .SH "NOTES" .IX Header "NOTES" .SS "Initializing parameter records." .IX Subsection "Initializing parameter records." The parameter record \fIcontrol\fR should always be initialized from supplied default records: .PP .Vb 1 \& lm_control_struct control = lm_control_double; /* or _float */ .Ve .PP After this, parameters may be overwritten: .PP .Vb 2 \& control.patience = 500; /* allow more iterations */ \& control.verbosity = 15; /* for verbose monitoring */ .Ve .PP An application written this way is guaranteed to work even if new parameters are added to \fIlm_control_struct\fR. .PP Conversely, addition of parameters is not considered an \s-1API\s0 change; it may happen without increment of the major version number. .SH "EXAMPLES" .IX Header "EXAMPLES" .SS "Fitting a surface" .IX Subsection "Fitting a surface" Fit a data set y(t) by a function f(t;p) where t is a two-dimensional vector: .PP .Vb 2 \& #include "lmmin.h" \& #include \& \& /* 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.