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 | b). Geometric and computational properties of the method are analyzed in detail and illustrated by numerical examples.

A Gauss-Newton iteration for Total Least Squares problems

Fasino, Dario
;
2018

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 | b). Geometric and computational properties of the method are analyzed in detail and illustrated by numerical examples.
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11390/1134964
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