man/lmcurve.3

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.\" ========================================================================
.\"
.IX Title "lmcurve 3"
.TH lmcurve 3 "2015-11-27" "perl v5.20.2" "lmfit manual"
.\" For nroff, turn off justification.  Always turn off hyphenation; it makes
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.nh
.SH "NAME"
lmcurve \- Levenberg\-Marquardt least\-squares fit of a curve (t,y)
.SH "SYNOPSIS"
.IX Header "SYNOPSIS"
\&\fB#include <lmcurve.h\fR>
.PP
\&\fBvoid lmcurve( const int\fR \fIn_par\fR\fB, double *\fR\fIpar\fR\fB, const int\fR \fIm_dat\fR\fB,
              const\ double *\fR\fIt\fR\fB, const\ double *\fR\fIy\fR\fB,
              double (*\fR\fIf\fR\fB)( const double \fR\fIti\fR\fB, const double *\fR\fIpar\fR\fB ),
              const\ lm_control_struct *\fR\fIcontrol\fR\fB,
              lm_status_struct *\fR\fIstatus\fR\fB);\fR
.PP
\&\fBvoid lmcurve_tyd(
              const int\fR \fIn_par\fR\fB, double *\fR\fIpar\fR\fB, const int\fR \fIm_dat\fR\fB,
              const\ double *\fR\fIt\fR\fB, const\ double *\fR\fIy\fR\fB, const\ double *\fR\fIdy\fR\fB,
              double (*\fR\fIf\fR\fB)( const double \fR\fIti\fR\fB, const double *\fR\fIpar\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(BIlmcurve()\fB\fR and \fB\f(BIlmcurve_tyd()\fB\fR wrap the more generic minimization function \fB\f(BIlmmin()\fB\fR, for use in curve fitting.
.PP
\&\fB\f(BIlmcurve()\fB\fR determines a vector \fIpar\fR that minimizes the sum of squared elements of a residue vector \fIr\fR[i] := \fIy\fR[i] \- \fIf\fR(\fIt\fR[i];\fIpar\fR). Typically, \fB\f(BIlmcurve()\fB\fR is used to approximate a data set \fIt\fR,\fIy\fR by a parametric function \fIf\fR(\fIti\fR;\fIpar\fR). On success, \fIpar\fR represents a local minimum, not necessarily a global one; it may depend on its starting value.
.PP
\&\fB\f(BIlmcurve_tyd()\fB\fR does the same for a data set \fIt\fR,\fIy\fR,\fIdy\fR, where \fIdy\fR represents the standard deviation of empirical data \fIy\fR. Residues are computed as \fIr\fR[i] := (\fIy\fR[i] \- \fIf\fR(\fIt\fR[i];\fIpar\fR))/\fIdy\fR[i]. Users must ensure that all \fIdy\fR[i] are positive.
.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 ||\fIr\fR||.
.IP "\fIm_dat\fR" 4
.IX Item "m_dat"
Number of data points.
Length of vectors \fIt\fR and \fIy\fR.
Must statisfy \fIn_par\fR <= \fIm_dat\fR.
.IP "\fIt\fR" 4
.IX Item "t"
Array of length \fIm_dat\fR.
Contains the abcissae (time, or \*(L"x\*(R") for which function \fIf\fR will be evaluated.
.IP "\fIy\fR" 4
.IX Item "y"
Array of length \fIm_dat\fR.
Contains the ordinate values that shall be fitted.
.IP "\fIdy\fR" 4
.IX Item "dy"
Only in \fB\f(BIlmcurve_tyd()\fB\fR.
Array of length \fIm_dat\fR.
Contains the standard deviations of the values \fIy\fR.
.IP "\fIf\fR" 4
.IX Item "f"
A user-supplied parametric function \fIf\fR(ti;\fIpar\fR).
.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.
Parameters are explained in \fB\f(BIlmmin\fB\|(3)\fR.
.IP "\fIstatus\fR" 4
.IX Item "status"
A record used to return information about the minimization process:
For details, see \fB\f(BIlmmin\fB\|(3)\fR.
.SH "EXAMPLE"
.IX Header "EXAMPLE"
Fit a data set y(x) by a curve f(x;p):
.PP
.Vb 2
\&    #include "lmcurve.h"
\&    #include <stdio.h>
\&
\&    /* model function: a parabola */
\&
\&    double f( double t, const double *p )
\&    {
\&        return p[0] + p[1]*t + p[2]*t*t;
\&    }
\&
\&    int main()
\&    {
\&        int n = 3; /* number of parameters in model function f */
\&        double par[3] = { 100, 0, \-10 }; /* really bad starting value */
\&
\&        /* data points: a slightly distorted standard parabola */
\&        int m = 9;
\&        int i;
\&        double t[9] = { \-4., \-3., \-2., \-1.,  0., 1.,  2.,  3.,  4. };
\&        double y[9] = { 16.6, 9.9, 4.4, 1.1, 0., 1.1, 4.2, 9.3, 16.4 };
\&
\&        lm_control_struct control = lm_control_double;
\&        lm_status_struct status;
\&        control.verbosity = 7;
\&
\&        printf( "Fitting ...\en" );
\&        lmcurve( n, par, m, t, y, f, &control, &status );
\&
\&        printf( "Results:\en" );
\&        printf( "status after %d function evaluations:\en  %s\en",
\&                status.nfev, lm_infmsg[status.outcome] );
\&
\&        printf("obtained parameters:\en");
\&        for ( i = 0; i < n; ++i)
\&            printf("  par[%i] = %12g\en", i, par[i]);
\&        printf("obtained norm:\en  %12g\en", status.fnorm );
\&
\&        printf("fitting data as follows:\en");
\&        for ( i = 0; i < m; ++i)
\&            printf( "  t[%2d]=%4g y=%6g fit=%10g residue=%12g\en",
\&                    i, t[i], y[i], f(t[i],par), y[i] \- f(t[i],par) );
\&
\&        return 0;
\&    }
.Ve
.SH "COPYING"
.IX Header "COPYING"
Copyright (C) 2009\-2015 Joachim Wuttke, Forschungszentrum Juelich GmbH
.PP
Software: FreeBSD License
.PP
Documentation: Creative Commons Attribution Share Alike
.SH "SEE ALSO"
.IX Header "SEE ALSO"
\fBlmmin\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>.