The Use of Restarted Krylov Methods for Large Sparse Least Squares Problems

The use of restarted Krylov methods for solving large sparse linear least squares problems suffers from certain inherited difficulties. One of them arises in ill‐conditioned problems, when part of the residual vector is spanned by singular vectors that correspond to small singular values. The modification proposed in this article enables the Krylov subspaces to collect information on these vectors. This is achieved by combining the restarted Krylov process with the Heart iteration for computing small singular values. Numerical experiments illustrate the usefulness of the proposed approach.