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Methods for large scale total least squares problems. (English) Zbl 0974.65037
Authors’ abstract: The solution of the total least squares (TLS) problems, \[ \min_{E,f}\|(E, f)\|_F\quad\text{subject to }(A+E)x= b+f, \] can in the generic case be obtained from the right singular vector corresponding to the smallest singular value \(\sigma_{n+1}\) of \((A,b)\). When \(A\) is large and sparse (or structured) a method bases on Rayleigh quotient iteration (RQI) has been suggested by A. Björck [Proc. 2nd Int. Workshop on Total Least Squares, Leuven 1996, SIAM 149-160 (1997; Zbl 0879.65105)]. In this method the problem is reduced to the solution of a sequence of symmetric, positive definite linear systems of the form \((A^TA- \overline\partial^2 I)z= g\), where \(\overline\sigma\) is an approximation to \(\sigma_{n+1}\). These linear systems are then solved by a preconditioned conjugate gradient method (PCGTLS). For TLS problems where \(A\) is large and sparse a (possibly incomplete) Cholesky factor of \(A^TA\) can usually be computed, and this provides a very efficient preconditioner. The resulting method can be used to solve a much wider range of problems than it is possible to solve by using Lanczos-type algorithms directly for the singular value problem.
In this paper the RQI-PCGTLS methods are further developed, and the choice of initial approximation and termination criteria are discussed. Numerical results confirm that the given algorithm achieves rapid convergence and good accuracy.

65F20 Numerical solutions to overdetermined systems, pseudoinverses
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