Preconditioning for nonsymmetry and time-dependence

McDonald, Eleanor and Hon, Sean and Pestana, Jennifer and Wathen, Andrew (2017) Preconditioning for nonsymmetry and time-dependence. In: 23rd International Conference on Domain Decomposition Methods, 2015-07-06 - 2015-07-10. (

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In this short paper, we decribe at least one simple and frequently arising situation |that of nonsymmetric real Toeplitz (constant diagonal) matrices| where we can guarantee rapid convergence of the appropriate iterative method by manipulating the problem into a symmetric form without recourse to the normal equations. This trick can be applied regardless of the nonnormality of the Toeplitz matrix. We also propose a symmetric and positive definite preconditioner for this situation which is proved to cluster eigenvalues and is by consequence guaranteed to ensure convergence in a number of iterations independent of the matrix dimension.