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BIT Numerical Mathematics

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29-07-2017

A Gauss–Newton iteration for Total Least Squares problems

Authors: Dario Fasino, Antonio Fazzi

Published in: BIT Numerical Mathematics | Issue 2/2018

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Abstract

The Total Least Squares solution of an overdetermined, approximate linear equation \(Ax \approx b\) minimizes a nonlinear function which characterizes the backward error. We devise a variant of the Gauss–Newton iteration with guaranteed convergence to that solution, under classical well-posedness hypotheses. At each iteration, the proposed method requires the solution of an ordinary least squares problem where the matrix A is modified by a rank-one term. In exact arithmetics, the method is equivalent to an inverse power iteration to compute the smallest singular value of the complete matrix \((A\mid b)\). Geometric and computational properties of the method are analyzed in detail and illustrated by numerical examples.

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We are grateful to an anonymous referee for having brought this citation to our attention.

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Title: A Gauss–Newton iteration for Total Least Squares problems
Authors: Dario Fasino
Antonio Fazzi
Publication date: 29-07-2017
Publisher: Springer Netherlands
Published in: BIT Numerical Mathematics / Issue 2/2018
Print ISSN: 0006-3835
Electronic ISSN: 1572-9125
DOI: https://doi.org/10.1007/s10543-017-0678-5

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