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Function: LSQFITD

Purpose: LSQFITD is a routine for making a least-squares fit of a
function to a set of data points. The method used is
described in: Marquardt, J.Soc.Ind.Appl.Math. 11, 431 (1963).
This method is a mixture of the steepest descent method and
the Taylor method.

Category: MATH

File: lsqfitd.c

Author: K.G. Begeman

Use: INTEGER LSQFITD( XDAT , Input DOUBLE PRECISION ARRAY
XDIM , Input INTEGER
YDAT , Input DOUBLE PRECISION ARRAY
WDAT , Input DOUBLE PRECISION ARRAY
NDAT , Input INTEGER
FPAR , In/Output DOUBLE PRECISION ARRAY
EPAR , Output DOUBLE PRECISION ARRAY
MPAR , Input INTEGER ARRAY
NPAR , Input INTEGER
TOL , Input DOUBLE PRECISION
ITS , Input INTEGER
LAB , Input DOUBLE PRECISION
FOPT ) Input INTEGER

LSQFITD Returns number of iterations needed to achieve
convergence according to TOL. When this
number is negative, the fitting was not
continued because a fatal error occurred:
-1 Too many free parameters, maximum is 32.
-2 No free parameters.
-3 Not enough degrees of freedom.
-4 Maximum number of iterations too small to
obtain a solution which satisfies TOL.
-5 Diagonal of matrix contains elements which
are zero.
-6 Determinant of the coefficient matrix is zero.
-7 Square root of negative number.
XDAT Contains coordinates of data points.
XDAT is two-dimensional: XDAT(XDIM,NDAT)
XDIM Dimension of fit.
YDAT Contains data points.
WDAT Contains weigths for data points.
NDAT Number of data points.
FPAR On input contains initial estimates of the
parameters for non-linear fits, on output the
fitted parameters.
EPAR Contains estimates of errors in fitted
parameters.
MPAR Logical mask telling which parameters are
free (MPAR(J)=non-zero) and which parameters
are fixed (MPAR(J)=0).
NPAR Number of parameters (free+fixed).
TOL Relative tolerance. LSQFITD stops when
successive iterations fail to produce a
decrement in reduced chi-squared less than
TOL. If TOL is less than the minimum tolerance
possible, TOL will be set to this value. This
means that maximum accuracy can be obtained by
setting TOL=0.0.
ITS Maximum number of iterations.
LAB Mixing parameter, LAB determines the initial
weight of steepest descent method relative to
the Taylor method. LAB should be a small
value (i.e. 0.01). LAB can only be zero when
the partial derivatives are independent of
the parameters. In fact in this case LAB
should be exactly equal to zero.
FOPT A value which is passed unmodified to FUNC
and DERV (see below).

Notes: The following routines have to be defined by the user:

DOUBLE PRECISION FUNCD( XDAT , Input DOUBLE PRECISION ARRAY
FPAR , Input DOUBLE PRECISION ARRAY
NPAR , Input INTEGER
FOPT ) Input INTEGER

FUNC Returns the function value of the function to
be fitted.
XDAT Coordinate(s) of data point.
FPAR Parameter list.
NPAR Number of parameters.
FOPT A user defined option.

CALL DERVD( XDAT , Input DOUBLE PRECISION ARRAY
FPAR , Input DOUBLE PRECISION ARRAY
DPAR , Output DOUBLE PRECISION ARRAY
NPAR , Input INTEGER
FOPT ) Input INTEGER

XDAT Coordinate(s) of data point.
FPAR Parameter list.
EPAR Partial derivatives to the parameters of the
function to be fitted.
NPAR Number of parameters.
FOPT A user defined option.

In FORTRAN applications you need to specify the
following f2cvv syntax for the C to Fortran
interface somewhere in your source code:

C@ double precision function funcd( double precision,
C@ double precision, integer, integer )
C@ subroutine dervd( double precision, double precision,
C@ double precision, integer, integer )

Example: Fitting y(x) = a + b * x

DOUBLE PRECISION FUNCTION FUNCD( XDAT, FPAR, NPAR, FOPT )
...
FUNCD = FPAR(1) + FPAR(2) * XDAT
RETURN
END

SUBROUTINE DERVD( XDAT, FPAR, DPAR, NPAR, FOPT )
...
DPAR(1) = 1.0D0
DPAR(2) = XDAT
RETURN
END

Updates: May 7, 1990: KGB, Document created.
May 14, 1990: MXV, Document refereed.
Apr 13, 1993: KGB, Double precision version created.