Spectral Properties of Preconditioned Rational Toeplitz Matrices: The Nonsymmetric Case

Various preconditioners for symmetric positive-definite (SPD) Toeplitz matrices in circulant matrix form have recently been proposed. The spectral properties of the preconditioned SPD Toeplitz matrices have also been studied. In this research, Strang’s preconditioner $S_N $ and our preconditioner $K_N $ are applied to an $N \times N$ nonsymmetric (or nonhermitian) Toeplitz system $T_N {\bf x} = {\bf b}$. For a large class of Toeplitz matrices, it is proved that the singular values of $S_N^{ - 1} T_N $ and $K_N^{ - 1} T_N $ are clustered around unity except for a fixed number independent of N. If $T_N $ is additionally generated by a rational function, the eigenvalues of $S_N^{ - 1} T_N $ and $K_N^{ - 1} T_N $ can be characterized directly. Let the eigenvalues of $S_N^{ - 1} T_N $ and $K_N^{ - 1} T_N $ be classified into the outliers and the clustered eigenvalues depending on whether they converge to 1 asymptotically. Then, the number of outliers depends on the order of the rational generating function, and the clustering radius is proportional to the magnitude of the last elements in the generating sequence used to construct the preconditioner. Numerical experiments are provided to illustrate our theoretical study.