A geometric view of Krylov subspace methods on singular systems |
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| Authors: | Ken Hayami Masaaki Sugihara |
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| Institution: | 1. National Institute of Informatics, 2‐1‐2, Hitotsubashi, Chiyoda‐ku, Tokyo 101‐8430, Japan;2. The Graduate University for Advanced Studies (Sokendai), 2‐1‐2, Hitotsubashi, Chiyoda‐ku, Tokyo 101‐8430, Japan;3. Department of Mathematical Informatics, Graduate School of Information Science and Technology, The University of Tokyo, 7‐3‐1, Hongo, Bunkyo‐ku, Tokyo 113‐8656, Japan |
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| Abstract: | We give a geometric framework for analysing iterative methods on singular linear systems A x = b and apply them to Krylov subspace methods. The idea is to decompose the method into the ?(A) component and its orthogonal complement ?(A)?, where ?(A) is the range of A. We apply the framework to GMRES, GMRES(k) and GCR(k), and derive conditions for convergence without breakdown for inconsistent and consistent singular systems. The approach also gives a geometric interpretation and different proofs of the conditions obtained by Brown and Walker for GMRES. We also give examples arising in the finite difference discretization of two‐point boundary value problems of an ordinary differential equation. Copyright © 2010 John Wiley & Sons, Ltd. |
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| Keywords: | Krylov subspace method GMRES method GCR(k) method singular systems least squares problems |
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