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, an...