Optimal preconditioners for systems defined by functions of Toeplitz matrices

Abstract We propose several circulant preconditioners for systems defined by some functions g of Toeplitz matrices A n . In this paper we are interested in solving g ( A n ) x = b by the preconditioned conjugate method or the preconditioned minimal residual method, namely in the cases when g ( z ) are the functions e z , sin ⁡ z and cos ⁡ z . Numerical results are given to show the effectiveness of the proposed preconditioners.

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