casacore
scimath
Fitting.h
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//# Fitting.h: Module for various forms of mathematical fitting
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//# Copyright (C) 1995,1999-2002,2004
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//# Associated Universities, Inc. Washington DC, USA.
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//#
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//# This library is free software; you can redistribute it and/or modify it
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//# under the terms of the GNU Library General Public License as published by
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//# the Free Software Foundation; either version 2 of the License, or (at your
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//# option) any later version.
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//#
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//# This library is distributed in the hope that it will be useful, but WITHOUT
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//# ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
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//# FITNESS FOR A PARTICULAR PURPOSE. See the GNU Library General Public
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//# License for more details.
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//#
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//# You should have received a copy of the GNU Library General Public License
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//# along with this library; if not, write to the Free Software Foundation,
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//# Inc., 675 Massachusetts Ave, Cambridge, MA 02139, USA.
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//#
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//# Correspondence concerning AIPS++ should be addressed as follows:
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//# Internet email: aips2-request@nrao.edu.
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//# Postal address: AIPS++ Project Office
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//# National Radio Astronomy Observatory
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//# 520 Edgemont Road
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//# Charlottesville, VA 22903-2475 USA
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//#
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//# $Id$
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#ifndef SCIMATH_FITTING_H
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#define SCIMATH_FITTING_H
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#include <casacore/casa/aips.h>
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#include <casacore/scimath/Fitting/LSQFit.h>
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#include <casacore/scimath/Fitting/LinearFit.h>
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#include <casacore/scimath/Fitting/LinearFitSVD.h>
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#include <casacore/scimath/Fitting/NonLinearFit.h>
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#include <casacore/scimath/Fitting/NonLinearFitLM.h>
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namespace
casacore
{
//# NAMESPACE CASACORE - BEGIN
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// <module>
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//
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// <summary>
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// Module for various forms of mathematical fitting.
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// </summary>
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//
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// <prerequisite>
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// <li> Basic principles can be found in
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// <a href="../notes/224.html">Note 224</a>.
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// </prerequisite>
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//
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// <reviewed reviewer="Neil Killeen" date="2000/06/01"
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// demos="dLSQFit,dConstraints">
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// </reviewed>
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//
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// <synopsis>
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//
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// The Fitting module holds various classes and functions related
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// to fitting models to data. Currently only least-squares fits
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// are handled.
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//
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// <H3>Least-Squares Fits</H3>
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//
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// We are given N data points, which
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// we will fit to a function with M adjustable parameters.
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// N should normally be greater than M, and at least M non-dependent relations
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// between the parameters should be given. In cases where there are less than
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// M independent points, Singular-Value-Deconvolution methods are available.
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// Each condition equation can be given an
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// (estimated) standard deviation, which is comparable to the statistical
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// weight, which is often used in place of the standard deviation.
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//
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// The best fit is assumed to be the one which minimises the 'chi-squared'.
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//
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// In the (rather common) case that individual errors are not known for
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// the individual data points, one can <em>assume</em> that the
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// individual errors are unity, calculate the best fit function, and then
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// <em>estimate</em> the errors (assuming they are all identical) by
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// inverting the <em>normal equations</em>.
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// Of course, in this case we do not have an independent estimate of
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// chi<sup>2</sup>.
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//
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// The methods used in the Fitting module are described in
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// <a href="../notes/224.html">Note 224</a>.
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// The methods (both standard and
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// SVD) are based on a Cholesky decomposition of the normal equations.
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//
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// General background can also be found in <EM>Numerical Recipes</EM> by
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// Press <EM>et al.</EM>.
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//
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// <H3>Linear Least-Squares Fits</H3>
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//
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// The <em>linear least squares</em> solution assumes that the fit function
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// is a linear combination of M linear condition equations.
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// It is important to note that <em>linear</em> refers to the dependence on
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// the parameters; the condition equations may be very non-linear in the
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// dependent arguments.
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//
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// The linear least squares problem is solved by explicitly
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// forming and inverting the normal equations.
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// If the normal equations are close to
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// singular, the <em>singular value decomposition</em> (SVD) method may be
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// used. <em>Numerical Recipes</em> suggests the SVD be always used, however
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// this advice is not universally accepted.
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//
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// <H3>Linear Least-Squares Fits with Known Linear Constraints</H3>
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//
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// Sometimes there are not enough independent observations, <EM>i.e.</EM>, the
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// number of data points <I>N</I> is less than the number of adjustable
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// parameters <I>M</I>. In this case the least-squares problem cannot be
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// solved unless additional ``constraints'' on the adjustable parameters can
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// be introduced. Under other circumstances, we may want to introduce
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// constraints on the adjustable
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// parameters to add additional information, <EM>e.g.</EM>, the sum of angles
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// of a triangle. In more complex cases, the forms of the constraints
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// are unknown. Here we confine ourselves to
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// least-squares fit problems in which the forms of constraints are known.
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//
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// If the forms of constraint equations are known, the least-squares
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// problem can be solved. (In the case where not
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// enough independent observations are available, a minimum number of
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// sufficient constraint equations have to be provided. The singular value
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// decomposition method can be used to calculate the minimum number of
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// orthogonal constraints needed).
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//
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// <H3>Nonlinear Least-Squares Fits</H3>
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//
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// We now consider the situation where the fitted function
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// depends <EM>nonlinearly</EM> on the set of
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// <I>M</I> adjustable parameters.
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// But with nonlinear dependences the minimisation of chi<sup>2</sup> cannot
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// proceed as in the linear case.
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// However, we can <EM>linearise</EM> the problem, find an
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// approximate solution, and then iteratively seek the minimising solution.
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// The iteration stops when e.g. the adjusted parameters do not change
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// anymore. In general it is very difficult to find a general solution that
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// finds a global minimum, and the solution has to be matched with the problem.
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// The Levenberg-Marquardt algorithm is a general non-linear fitting method
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// which can produce correct results in many cases. It has been included, but
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// always be aware of possible problems with non-linear solutions.
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//
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// <H3>What Is Available?</H3>
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//
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// The basic classes are <linkto class=LSQFit>LSQFit</linkto> and
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// <linkto class=LSQaips>LSQaips</linkto>.
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// They provide the basic framework for normal equation generation, solving
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// the normal equations and iterating in the case of non-linear equations.
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//
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// The <em>LSQFit</em> class uses a native C++ interface (pointers and
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// iterators). They handle real data and complex data.
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// The <em>LSQaips</em> class offers the functionality of <em>LSQFit</em>,
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// but with an additional Casacore Array interface.<br>
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//
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// Functionality is
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// <ol>
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// <li> Fit a linear combination of functions to data points, and, optionally,
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// use supplied constraint conditions on the adjustable parameters.
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// <li> Fit a nonlinear function to data points. The adjustable parameters
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// are parameters inside the function.
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// <li> Repetitively perform a linear fit for every line of pixels parallel
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// to any axis in a Lattice.
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// <li> Solve (as opposed to fit to a set of data), a set of equations
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// </ol>
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//
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// In addition to the basic Least Squares routines in the <src>LSQFit</src> and
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// <src>LSQaips</src> classes, this module contains also a set of direct
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// data fitters:
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// <ul>
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// <li> <src>Fit2D</src>
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// <li> <src>LinearFit</src>
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// <li> <src>LinearFitSVD</src>
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// <li> <src>NonLinearFit</src>
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// <li> <src>NonLinearFitLM</src>
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// </ul>
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// Furthermore class <src>LatticeFit</src> can do fitting on lattices.
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//
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// Note that the basic functions have <em>LSQ</em> in their title; the
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// one-step fitting functions <em>Fit</em>.
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//
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// The above fitting problems can usually be solved by directly
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// calling the <src>fit()</src> member function provided by one of the
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// <src>Fit</src> classes above, or by
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// gradually building the normal equation matrix and solving the
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// normal equations (<src>solve()</src>).
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//
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// A Distributed Object interface to the classes is available
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// (<src>DOfitting</src>) for use in the <I>Glish</I> <src>dfit</src>
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// object, available through the <src>fitting.g</src> script.
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//
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// </synopsis>
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//
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// <motivation>
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// This module was motivated by baseline subtraction/continuum fitting in the
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// first instance.
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// </motivation>
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//
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// <todo asof="2004/09/22">
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// <li> extend the Fit interface classes to cater for building the
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// normal equations in parts.
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// </todo>
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//
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// </module>
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//
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}
//# NAMESPACE CASACORE - END
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#endif
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casacore
this file contains all the compiler specific defines
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