Preconditioned Krylov subspace method for the solution of least‐squares problems

Abstract

AbstractWe consider preconditioned Krylov subspace iteration methods, e.g., CG, LSQR and GMRES, for the solution of large sparse least‐squares problems min ∥Ax – b ∥2, with A ∈ Rm×n, based on the Krylov subspaces Kk (BA, Br) and Kk (AB, r), respectively, where B ∈ Rn×m is the preconditioning matrix. More concretely, we propose and implement a class of incomplete QR factorization preconditioners based on the Givens rotations and analyze in detail the efficiency and robustness of the correspondingly preconditioned Krylov subspace iteration methods. A number of numerical experiments are used to further examine their numerical behaviour. It is shown that for both overdetermined and underdetermined least‐squares problems, the preconditioned GMRES methods are the best for large, sparse and ill‐conditioned matrices in terms of both CPU time and iteration step. Also, comparisons with the diagonal scaling and the RIF preconditioners are given to show the superiority of the newly‐proposed GMRES‐type methods. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)