Vector Extrapolation Applied to Truncated Singular Value Decomposition and Truncated Iteration

This paper is concerned with the computation of accurate approximate solutions of linear systems of equations and linear least-squares problems with a very ill-conditioned matrix and error-contaminated data. The solution of this kind of problems requires regularization. Common regularization methods include the truncated singular value decomposition and truncated iteration with a Krylov subspace method. It can be difficult to determine when to truncate. Recently, it has been demonstrated that extrapolation of approximate solutions determined by truncated singular value decomposition gives a new sequence of approximate solutions that is less sensitive to the error in the data than the original approximate solutions. The present paper describes a novel approach to determine a suitable truncation index by comparing the original and extrapolated approximate solutions. Applications to truncated singular value decomposition and the LSQR iterative method are presented.