A Minimum-Phase LU Factorization Preconditioner for Toeplitz Matrices

Abstract

A new preconditioner is proposed for the solution of an $N \times N$ Toeplitz system $T_N {\bf x} = {\bf b}$, where $T_N $ can be symmetric indefinite or nonsymmetric, by preconditioned iterative methods. The preconditioner $F_N $ is obtained based on factorizing the generating function $T(z)$ into the product of two terms corresponding, respectively, to minimum-phase causal and anticausal systems and is therefore called the minimum-phase LU (MPLU) factorization preconditioner. Due to the minimum-phase property, $||F_N^{ - 1} ||$ is bounded. For rational Toeplitz matrices $T_N $ with generating function $T(z) = A(z^{ - 1}/Bz^{ - 1} ) + C(z)/D(z)$, where $A(z)$, $B(z)$, $C(z)$, and $D(z)$ are polynomials of orders $p_1 $, $q_1 $, $p_2 $, and $q_2 $, it is shown that the eigenvalues of $F_N^{ - 1} T_N $ are repeated exactly at 1 except at most $\alpha _F $ outliers, where $\alpha _F $ depends on $p_1 $, $q_1 $, $p_2 $, $q_2 $, and the number w of the zeros of $\tilde T(z) = A(z^{ - 1} )D(z) + B(z^{ - 1} )C(z)$ outside the unit circle. A preconditioner $K_N $ in circulant form generalized from the symmetric case is also presented for comparison.