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394 lines
13 KiB
394 lines
13 KiB
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.\" ========================================================================
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.\"
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.IX Title "lmmin 3"
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.TH lmmin 3 "2015-11-27" "perl v5.20.2" "lmfit manual"
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.\" For nroff, turn off justification. Always turn off hyphenation; it makes
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.\" way too many mistakes in technical documents.
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.if n .ad l
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.nh
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.SH "NAME"
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lmmin \- Levenberg\-Marquardt least\-squares minimization
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.SH "SYNOPSIS"
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.IX Header "SYNOPSIS"
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\&\fB#include <lmmin.h\fR>
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.PP
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\&\fBvoid lmmin( const int\fR \fIn_par\fR\fB, double *\fR\fIpar\fR\fB, const int\fR \fIm_dat\fR\fB,
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const\ void *\fR\fIdata\fR\fB,
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void *\fR\fIevaluate\fR\fB(
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const\ double *\fR\fIpar\fR\fB, const int \fR\fIm_dat\fR\fB,
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const\ void *\fR\fIdata\fR\fB, double *\fR\fIfvec\fR\fB, int *\fR\fIuserbreak\fR\fB),
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const\ lm_control_struct *\fR\fIcontrol\fR\fB,
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lm_status_struct *\fR\fIstatus\fR\fB );\fR
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.PP
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\&\fBextern const lm_control_struct lm_control_double;\fR
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.PP
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\&\fBextern const lm_control_struct lm_control_float;\fR
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.PP
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\&\fBextern const char *lm_infmsg[];\fR
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.PP
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\&\fBextern const char *lm_shortmsg[];\fR
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.SH "DESCRIPTION"
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.IX Header "DESCRIPTION"
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\&\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().
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On success, \fIpar\fR represents a local minimum, not necessarily a global one; it may depend on its starting value.
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.PP
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For applications in curve fitting, the wrapper function \fB\f(BIlmcurve\fB\|(3)\fR offers a simplified \s-1API.\s0
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.PP
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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.
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.PP
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Function arguments:
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.IP "\fIn_par\fR" 4
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.IX Item "n_par"
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Number of free variables.
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Length of parameter vector \fIpar\fR.
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.IP "\fIpar\fR" 4
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.IX Item "par"
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Parameter vector.
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On input, it must contain a reasonable guess.
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On output, it contains the solution found to minimize ||\fIfvec\fR||.
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.IP "\fIm_dat\fR" 4
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.IX Item "m_dat"
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Length of vector \fIfvec\fR.
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Must statisfy \fIn_par\fR <= \fIm_dat\fR.
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.IP "\fIdata\fR" 4
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.IX Item "data"
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This pointer is ignored by the fit algorithm,
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except for appearing as an argument in all calls to the user-supplied
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routine \fIevaluate\fR.
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.IP "\fIevaluate\fR" 4
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.IX Item "evaluate"
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Pointer to a user-supplied function that computes \fIm_dat\fR elements of vector \fIfvec\fR for a given parameter vector \fIpar\fR.
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If \fIevaluate\fR return with *\fIuserbreak\fR set to a negative value, \fB\f(BIlmmin()\fB\fR will interrupt the fitting and terminate.
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.IP "\fIcontrol\fR" 4
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.IX Item "control"
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Parameter collection for tuning the fit procedure.
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In most cases, the default &\fIlm_control_double\fR is adequate.
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If \fIf\fR is only computed with single-precision accuracy,
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\&\fI&lm_control_float\fR should be used.
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See also below, \s-1NOTES\s0 on initializing parameter records.
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.Sp
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\&\fIcontrol\fR has the following members (for more details, see the source file \fIlmstruct.h\fR):
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.RS 4
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.IP "\fBdouble\fR \fIcontrol.ftol\fR" 4
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.IX Item "double control.ftol"
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Relative error desired in the sum of squares.
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Recommended setting: somewhat above machine precision; less if \fIfvec\fR is computed with reduced accuracy.
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.IP "\fBdouble\fR \fIcontrol.xtol\fR" 4
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.IX Item "double control.xtol"
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Relative error between last two approximations.
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Recommended setting: as \fIftol\fR.
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.IP "\fBdouble\fR \fIcontrol.gtol\fR" 4
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.IX Item "double control.gtol"
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A measure for degeneracy.
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Recommended setting: as \fIftol\fR.
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.IP "\fBdouble\fR \fIcontrol.epsilon\fR" 4
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.IX Item "double control.epsilon"
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Step used to calculate the Jacobian.
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Recommended setting: as \fIftol\fR, but definitely less than the accuracy of \fIfvec\fR.
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.IP "\fBdouble\fR \fIcontrol.stepbound\fR" 4
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.IX Item "double control.stepbound"
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Initial bound to steps in the outer loop, generally between 0.01 and 100; recommended value is 100.
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.IP "\fBint\fR \fIcontrol.patience\fR" 4
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.IX Item "int control.patience"
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Used to set the maximum number of function evaluations to patience*n_par.
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.IP "\fBint\fR \fIcontrol.scale_diag\fR" 4
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.IX Item "int control.scale_diag"
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Logical switch (0 or 1).
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If 1, then scale parameters to their initial value.
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This is the recommended setting.
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.IP "\fBFILE*\fR \fIcontrol.msgfile\fR" 4
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.IX Item "FILE* control.msgfile"
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Progress messages will be written to this file.
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Typically \fIstdout\fR or \fIstderr\fR.
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The value \fI\s-1NULL\s0\fR will be interpreted as \fIstdout\fR.
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.IP "\fBint\fR \fIcontrol.verbosity\fR" 4
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.IX Item "int control.verbosity"
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If nonzero, some progress information from within the \s-1LM\s0 algorithm
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is written to control.stream.
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.IP "\fBint\fR \fIcontrol.n_maxpri\fR" 4
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.IX Item "int control.n_maxpri"
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\&\-1, or maximum number of parameters to print.
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.IP "\fBint\fR \fIcontrol.m_maxpri\fR" 4
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.IX Item "int control.m_maxpri"
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\&\-1, or maximum number of residuals to print.
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.RE
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.RS 4
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.RE
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.IP "\fIstatus\fR" 4
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.IX Item "status"
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A record used to return information about the minimization process:
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.RS 4
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.IP "\fBdouble\fR \fIstatus.fnorm\fR" 4
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.IX Item "double status.fnorm"
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|
Norm of the vector \fIfvec\fR;
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.IP "\fBint\fR \fIstatus.nfev\fR" 4
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.IX Item "int status.nfev"
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Actual number of iterations;
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.IP "\fBint\fR \fIstatus.outcome\fR" 4
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.IX Item "int status.outcome"
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Status of minimization;
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for the corresponding text message, print \fIlm_infmsg\fR\fB[\fR\fIstatus.outcome\fR\fB]\fR;
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for a short code, print \fIlm_shortmsg\fR\fB[\fR\fIstatus.outcome\fR\fB]\fR.
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.IP "\fBint\fR \fIstatus.userbreak\fR" 4
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.IX Item "int status.userbreak"
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Set when termination has been forced by the user-supplied routine \fIevaluate\fR.
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.RE
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.RS 4
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.RE
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.SH "NOTES"
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.IX Header "NOTES"
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.SS "Initializing parameter records."
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.IX Subsection "Initializing parameter records."
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The parameter record \fIcontrol\fR should always be initialized
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from supplied default records:
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.PP
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.Vb 1
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\& lm_control_struct control = lm_control_double; /* or _float */
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.Ve
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.PP
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After this, parameters may be overwritten:
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.PP
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.Vb 2
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\& control.patience = 500; /* allow more iterations */
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\& control.verbosity = 15; /* for verbose monitoring */
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.Ve
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.PP
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An application written this way is guaranteed to work even if new parameters
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are added to \fIlm_control_struct\fR.
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.PP
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Conversely, addition of parameters is not considered an \s-1API\s0 change; it may happen without increment of the major version number.
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.SH "EXAMPLES"
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.IX Header "EXAMPLES"
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.SS "Fitting a surface"
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.IX Subsection "Fitting a surface"
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Fit a data set y(t) by a function f(t;p) where t is a two-dimensional vector:
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.PP
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.Vb 2
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\& #include "lmmin.h"
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\& #include <stdio.h>
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\&
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\& /* fit model: a plane p0 + p1*tx + p2*tz */
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\& double f( double tx, double tz, const double *p )
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\& {
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\& return p[0] + p[1]*tx + p[2]*tz;
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\& }
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\&
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\& /* data structure to transmit data arays and fit model */
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\& typedef struct {
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\& double *tx, *tz;
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\& double *y;
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\& double (*f)( double tx, double tz, const double *p );
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\& } data_struct;
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\&
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\& /* function evaluation, determination of residues */
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\& void evaluate_surface( const double *par, int m_dat,
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\& const void *data, double *fvec, int *userbreak )
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\& {
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\& /* for readability, explicit type conversion */
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\& data_struct *D;
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\& D = (data_struct*)data;
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\&
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\& int i;
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\& for ( i = 0; i < m_dat; i++ )
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\& fvec[i] = D\->y[i] \- D\->f( D\->tx[i], D\->tz[i], par );
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\& }
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\&
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\& int main()
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\& {
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\& /* parameter vector */
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\& int n_par = 3; /* number of parameters in model function f */
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\& double par[3] = { \-1, 0, 1 }; /* arbitrary starting value */
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\&
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\& /* data points */
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\& int m_dat = 4;
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\& double tx[4] = { \-1, \-1, 1, 1 };
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\& double tz[4] = { \-1, 1, \-1, 1 };
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\& double y[4] = { 0, 1, 1, 2 };
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\&
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\& data_struct data = { tx, tz, y, f };
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\&
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\& /* auxiliary parameters */
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\& lm_status_struct status;
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\& lm_control_struct control = lm_control_double;
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\& control.verbosity = 3;
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\&
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\& /* perform the fit */
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\& printf( "Fitting:\en" );
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\& lmmin( n_par, par, m_dat, (const void*) &data, evaluate_surface,
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\& &control, &status );
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\&
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\& /* print results */
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\& printf( "\enResults:\en" );
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\& printf( "status after %d function evaluations:\en %s\en",
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\& status.nfev, lm_infmsg[status.outcome] );
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\&
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\& printf("obtained parameters:\en");
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\& int i;
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\& for ( i=0; i<n_par; ++i )
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\& printf(" par[%i] = %12g\en", i, par[i]);
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\& printf("obtained norm:\en %12g\en", status.fnorm );
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\&
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\& printf("fitting data as follows:\en");
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\& double ff;
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\& for ( i=0; i<m_dat; ++i ){
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\& ff = f(tx[i], tz[i], par);
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\& printf( " t[%2d]=%12g,%12g y=%12g fit=%12g residue=%12g\en",
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\& i, tx[i], tz[i], y[i], ff, y[i] \- ff );
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\& }
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\&
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\& return 0;
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\& }
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.Ve
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.SS "More examples"
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.IX Subsection "More examples"
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For more examples, see the homepage and directories demo/ and test/ in the source distribution.
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.SH "COPYING"
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.IX Header "COPYING"
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Copyright (C):
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1980\-1999 University of Chicago
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2004\-2015 Joachim Wuttke, Forschungszentrum Juelich GmbH
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.PP
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Software: FreeBSD License
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.PP
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Documentation: Creative Commons Attribution Share Alike
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.SH "SEE ALSO"
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.IX Header "SEE ALSO"
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\fBlmcurve\fR(3)
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.PP
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Homepage: http://apps.jcns.fz\-juelich.de/lmfit
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.SH "BUGS"
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.IX Header "BUGS"
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Please send bug reports and suggestions to the author <j.wuttke@fz\-juelich.de>.
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