METHODS FOR LARGE SCALE TOTAL LEAST SQUARES PROBLEMS.

The solution of the total least squares (TLS) problems, min_E,f ¦ ¦(E,f)¦ ¦_F subject to (A + E)χ = b + f, can in the generic case be obtained from the right singular vector corresponding to the smallest singular value σ_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 Björck. In this method the problem is reduced to the solution of a sequence of symmetric, positive definite linear systems of the form (A^T A - &isgma;¯²I)z = g, where &isgma¯ is an approximation to σ_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^T A 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. [ABSTRACT FROM AUTHOR]