A Minimum-Phase LU Factorization Preconditioner for Toeplitz Matrices

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.