Methods for Large Scale Total Least Squares Problems

• Published in: SIAM journal on matrix analysis and applications Vol. 22; no. 2; pp. 413 - 429
• Main Authors: Björck, Åke, Heggernes, P., Matstoms, P.
• Format: Journal Article
• Language: English
• Published: Philadelphia Society for Industrial and Applied Mathematics 2001
• Subjects: Accuracy; Algorithms; Approximation; Conjugate gradient method; Iterative methods; Methods; NATURAL SCIENCES; NATURVETENSKAP; Rayleigh quotient iteration; Singular values; Sparsity; Total least squares
• ISSN: 0895-4798; 1095-7162; 1095-7162
• DOI: 10.1137/S0895479899355414

The solution of the total least squares (TLS) problems, $\min_{E,f}\|(E,f)\|_F$ 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 based on Rayleigh quotient iteration (RQI) has been suggested by Bjorck. In this method the problem is reduced to the solution of a sequence of symmetric, positive definite linear systems of the form $(A^TA-\bar\sigma^2I)z=g$, where $\bar\sigma$ is an approximation to $\sigma_{n+1}$. These linear systems are then solved by a {\em preconditioned} conjugate gradient method (PCGTLS). For TLS problems where A is large and sparse a (possibly incomplete) Cholesky factor of ATA 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 method is 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.}