NII Technical Report (NII-2013-004E)

Title Convergence of Inner-iteration GMRES Methods for Least Squares Problems (Revised Version)
Authors Keiichi Morikuni and Ken Hayami
Abstract We develop a general convergence theory for the generalized minimal residual method preconditioned by inner iterations for solving least squares problems. The inner iterations are performed by stationary iterative methods. We also present theoretical justifications for using specific inner iterations such as the Jacobi and SOR-type methods. The theory improves previous work [K. Morikuni and K. Hayami, SIAM J. Matrix Appl. Anal., 34 (2013), pp. 1–22], particularly in the rank-deficient case. We also characterize the spectrum of the preconditioned coefficient matrix by the spectral radius of the iteration matrix for the inner iterations, and give a convergence bound for the proposed methods. Finally, numerical experiments show that the proposed methods are more robust and efficient compared to previous methods for some rank-deficient problems.
Language English
Published Dec 3, 2013
Pages 24p
PDF File 13-004E.pdf

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National Institute of Informatics