NII Technical Report (NII-2004-006E)

Title Preconditioned GMRES Methods for Least Squares Problems
Authors Tokushi ITO and Ken HAYAMI
Abstract For least squares problems of minimizing || b - A x ||_2 where A is a large sparse m x n (m >= n) matrix, the common method is to apply the conjugate gradient method to the normal equation A^T A x = A^T b. However, the condition number of A^T A is square of that of A, and convergence becomes problematic for severely ill-conditioned problems even with preconditioning. In this paper, we propose two methods for applying the GMRES method to the least squares problem by using a n x m matrix B. We give the necessary and sufficient condition that B should satisfy in order that the proposed methods give a least squares solution. Then, for implementations for B, we propose an incomplete QR decomposition IMGS(l). Numerical experiments show that the simplest case l=0, which is equivalent to B= ( diag (A^T A) )^(-1) A^T, gives best results, and converges faster than previous methods for severely ill-conditioned problems.
Language English
Published May 7, 2004
Pages 29p
PDF File 04-006E.pdf

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National Institute of Informatics