<title>Minimum-phase LU factorization preconditioner for Toeplitz matrices</title>

Abstract

A new preconditioner is proposed for the solution of an N X N Toeplitz system T_Nx equals 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 therefore called the minimum-phase LU (MPLU) factorization preconditioner. Due to the minimum-phase property, (parallel)F_N^-1(parallel) is bounded. For rational Toeplitz T_N with generating function T(z) equals A(z^-1)/B(z^-1) + C(z)/D(z), where A(z), B(z), C(z), and D(z) are polynomials of orders p1, q1, p2, and q2, we show that the eigenvalues of F_N^-1T_N are repeated exactly at 1 except at most (alpha) F outliers, where (alpha) F depends on p1, q1, p2, q2, and the number (omega) of the roots of T(z) equals 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.