man/lmmin.3

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.\" ========================================================================
.\"
.IX Title "lmmin 3"
.TH lmmin 3 "2015-11-27" "perl v5.20.2" "lmfit manual"
.\" For nroff, turn off justification.  Always turn off hyphenation; it makes
.\" way too many mistakes in technical documents.
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.nh
.SH "NAME"
lmmin \- Levenberg\-Marquardt least\-squares minimization
.SH "SYNOPSIS"
.IX Header "SYNOPSIS"
\&\fB#include <lmmin.h\fR>
.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 <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
.SS "More examples"
.IX Subsection "More examples"
For more examples, see the homepage and directories demo/ and test/ in the source distribution.
.SH "COPYING"
.IX Header "COPYING"
Copyright (C):
   1980\-1999 University of Chicago
   2004\-2015 Joachim Wuttke, Forschungszentrum Juelich GmbH
.PP
Software: FreeBSD License
.PP
Documentation: Creative Commons Attribution Share Alike
.SH "SEE ALSO"
.IX Header "SEE ALSO"
\fBlmcurve\fR(3)
.PP
Homepage: http://apps.jcns.fz\-juelich.de/lmfit
.SH "BUGS"
.IX Header "BUGS"
Please send bug reports and suggestions to the author <j.wuttke@fz\-juelich.de>.