A full-Newton approach to separable nonlinear least squares problems and its application to discrete least squares rational approximation
Abstract
We consider a class of non-linear least squares problems that are widely used in fitting experimental
data. A defining characteristic of the models we will consider is that the solution parameters may be separated into
two classes, those that enter the problem linearly and those that enter non-linearly. Problems of this type are known as
separable non-linear least squares (SNLLS) problems and are often solved using a Gauss-Newton algorithm that was
developed in Golub and Pereyra [SIAM J. Numer. Anal., 10 (1973), pp. 413–432] and has been very widely applied.
We develop a full-Newton algorithm for solving this problem. Exploiting the structure of the general problem leads to
a surprisingly compact algorithm which exhibits all of the excellent characteristics of the full-Newton approach (e.g.
rapid convergence on problems with large residuals). Moreover, for certain problems of quite general interest, the
per iteration cost for the full-Newton algorithm compares quite favorably with that of the Gauss-Newton algorithm.
We explore one such problem, that of discrete least-squares fitting of rational functions.
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