A Minimum-Phase LU Factorization Preconditioner for Toeplitz Matrices

Abstract

A new preconditioner is proposed for the solution of an N x N Toeplitz system TNX = b, where TN can be symmetric indefinite or nonsymmetric, by preconditioned iterative methods. The preconditioner FN 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, IIFJ1II is bounded. For rational Toeplitz TN with generating function T(z) = A(z1)/B(z1) + C(z)/D(z), where A(z), B(z), C(z) and D(z) are polynomials of orders p , q , P2 and q , we show that the eigenvalues of FJ1 TN are repeated exactly at 1 except at most cF outliers, where cF depends on Pi ,qi , P2 , q2 and the number w of the roots of T(z) = A(z1)D(z) + B(z1)C(z) outside the unit circle. A preconditioner IN in circulant form generalized from the symmetric case is also presented for comparison.

Inverse product Toeplitz preconditioners for non-Hermitian Toeplitz systems

“…By exploiting the property that rational Toeplitz matrices can be transformed to banded ones (cf. (3.5)), an O(n) iterative solver for rational Toeplitz systems was derived in [18] under the following assumptions:…”

“…In the following, we briefly introduce the preconditioner for (3.6) proposed in [18], and discuss how to handle rational generating function with zeros and how to obtain (3.2) from (3.1).…”

“…It has been shown that Q n is well-conditioned [18,Theorem 1] and there are at most ρ eigenvalues of…”

“…Preconditioners for rational Toeplitz matrices have been proposed and analyzed, and more precise theoretical results have been obtained. The main result is that the eigenvalues of the preconditioned matrix are repeated exactly at 1 except for a fixed number (that only depends on the generating function) of outliers [8,[18][19][20]. Therefore, when Krylov type methods such as the PCG method and the GMRES method are applied to preconditioned rational Toeplitz systems, the number of iterations is independent of the matrix size.…”

“…We note that the total costs of the iterative solvers proposed in [8,19,20] are all O(n log n) operations. On the other hand, under the assumption that the rational generating function has no zeros, Ku and Kuo [18] proposed an O(n) iterative solver for rational Toeplitz systems. In Section 3, we modify the preconditioner proposed in [18] to get a new preconditioner that is welldefined for rational functions with zeros.…”