Optimal trigonometric preconditioners for nonsymmetric Toeplitz systems

Abstract This paper is concerned with the solution of systems of linear equations TNXN = bN, where ∗TN∗NϵN denotes a sequence of nonsingular nonsymmetric Toeplitz matrices arising from a generating function of the Wiener class. We present a technique for the fast construction of optimal trigonometric preconditioners MN = MN(T′NTN) of the corresponding normal equation which can be extended to Toeplitz least squares problems in a straightforward way. Moreover, we prove that the spectrum of the preconditioned matrix MN1T′NTN is clustered at 1 such that the PCG-method applied to the normal equation converges superlinearly. Numerical tests confirm the theoretical expectations.

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