Inverse product Toeplitz preconditioners for non-Hermitian Toeplitz systems

In this paper, we first propose product Toeplitz preconditioners (in an inverse form) for non-Hermitian Toeplitz matrices generated by functions with zeros. Our inverse product-type preconditioner is of the form \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$T_F T_L^{-1} T_U^{-1}$\end{document} where TF, TL, and TU are full, band lower triangular, and band upper triangular Toeplitz matrices, respectively. Our basic idea is to decompose the generating function properly such that all factors TF, TL, and TU of the preconditioner are as well-conditioned as possible. We prove that under certain conditions, the preconditioned matrix has eigenvalues and singular values clustered around 1. Then we use a similar idea to modify the preconditioner proposed in Ku and Kuo (SIAM J Sci Stat Comput 13:1470–1487, 1992) to handle the zeros in rational generating functions. Numerical results, including applications to the computation of the stationary probability distribution of Markovian queuing models with batch arrival, are given to illustrate the good performance of the proposed preconditioners.