NII Technical Report (NII-2005-015J)

Title The Solution of Least Squares Problems Using GMRES Methods
Authors Ken HAYAMI and Tokushi ITO
Abstract The most commonly used iterative method for solving large sparse least squares problems with an mn coefficient matrix A is the CGLS method, which applies the (preconditioned) conjugate gradient method to the normal equation. In this paper, we consider alternative methods using an nm matrix B to transform the problem to equivalent least squares problems with square coefficient matrices AB or BA, and then applying the Generalized Minimal Residual (GMRES) method, which is a robust Krylov subspace iterative method for solving systems of linear equations with nonsymmetric coefficient matrices. Next, we give a sufficeint condition concerning B for the proposed methods to give a least squares solution without breakdown for arbitrary right hand side b, for over-determined, under-determined and possibly rank-deficient problems. Then, as an example for B, we propose the IMGS(l) method, which is an incomplete QR decomposition. Finally, we show by numercial experiments on full-rank over-determined and under-determined problems that, for ill-conditioned problems, the proposed method using the IMGS(0) method, which is equivalent to diagonal scaling, gives a least squares solution faster than previous preconditioned CGLS methods.
Language Japanese
Published Oct 29, 2005
Pages 20p
PDF File 05-015J.pdf

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National Institute of Informatics