Circulant/Skewcirculant Matrices as Preconditioners for Hermitian Toeplitz Systems
نویسنده
چکیده
We study the solutions of Hermitian positive definite Toeplitz systems Tnx = b by the preconditioned conjugate gradient method. For preconditioner An the convergence rate is known to be governed by the distribution of the eigenvalues of the preconditioned matrix A−1 n Tn . New properties of the circulant preconditioners introduced by Strang, R. Chan, T. Chan, Szegö/Grenander and Tyrtyshnikov are derived concerning the positive definiteness of An and the spectrum of A −1 n Tn. Furthermore, we introduce a new class of Toeplitz matrices, similar to the Wiener class. For this class we consider new preconditioners as products of circulant and skewcirculant matrices Cn and Sn, that are best approximations of Tn in the Frobenius norm, and study the spectra of the preconditioned matrices.
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