A preconditioned MINRES method for nonsymmetric Toeplitz matrices

Pestana, Jennifer and Wathen, Andrew (2014) A preconditioned MINRES method for nonsymmetric Toeplitz matrices. [MIMS Preprint]

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Circulant preconditioning for symmetric Toeplitz linear systems is well-established; theoretical guarantees of fast convergence for the conjugate gradient method are descriptive of the convergence seen in computations. This has led to robust and highly efficient solvers based on use of the fast Fourier transform exactly as originally envisaged in Gil Strang's `Proposal for Toeplitz Matrix Calculations' (Studies in Applied Mathematics, 74, pp. 171-176, 1986.). For nonsymmetric systems, the lack of generally descriptive convergence theory for most iterative methods of Krylov type has provided a barrier to such a comprehensive guarantee, though several methods have been proposed and some analysis of performance with the normal equations is available. In this paper, by the simple device of reordering, we rigorously establish a circulant preconditioned short recurrence Krylov subspace iterative method of minimum residual type for nonsymmetric (and possibly highly nonnormal) Toeplitz systems. Convergence estimates similar to those in the symmetric case are established.

Item Type: MIMS Preprint
Uncontrolled Keywords: circulant preconditioner, MINRES, Toeplitz matrix
Subjects: MSC 2010, the AMS's Mathematics Subject Classification > 65 Numerical analysis
Depositing User: Dr Jennifer Pestana
Date Deposited: 27 Jun 2014
Last Modified: 08 Nov 2017 18:18
URI: https://eprints.maths.manchester.ac.uk/id/eprint/2150

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