Preconditioned Krylov subspace method for the solution of least-squares problems |
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Authors: | jun-Feng Yin Ken Hayami Zhong-Zhi Bai |
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Institution: | Department of Mathematics, Tongji University1239 Siping Road, Shanghai, P. R. China |
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Abstract: | We 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 ∈ R m×n, based on the Krylov subspaces Kk (BA, Br) and Kk (AB, r), respectively, where B ∈ R n×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) |
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