Abstract
When solving linear systems with nonsymmetric Toeplitz or multilevel Toeplitz matrices using Krylov subspace methods, the coefficient matrix may be symmetrized. The preconditioned MINRES method can then be applied to this symmetrized system, which allows rigorous upper bounds on the number of MINRES iterations to be obtained. However, effective preconditioners for symmetrized (multilevel) Toeplitz matrices are lacking. Here, we propose novel ideal preconditioners and investigate the spectra of the preconditioned matrices. We show how these preconditioners can be approximated and demonstrate their effectiveness via numerical experiments.
Highlights
Linear systems (1.1)Anx = b, where An \in \BbbR n\times n is a Toeplitz or multilevel Toeplitz matrix, and b \in \BbbR n arise in a range of applications
When An is Hermitian, the conjugate gradient method (CG) [18] and MINRES [29] can be applied, and their descriptive convergence rate bounds guide the construction of effective preconditioners [7, 27]
In [35] it was shown that absolute value circulant preconditioners, which we describe give fast convergence for many Toeplitz problems
Summary
Anx = b, where An \in \BbbR n\times n is a Toeplitz or multilevel Toeplitz matrix, and b \in \BbbR n arise in a range of applications. Preconditioned iterative methods are often used to solve systems of the form (1.1). Convergence rates of preconditioned iterative methods for nonsymmetric Toeplitz matrices are difficult to describe. We describe ideal preconditioners for symmetrized (multilevel) Toeplitz matrices and show how these can be effectively approximated.
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