Band-Times-Circulant Preconditioners for Non-Symmetric Real Toeplitz Systems with Unknown Generating Function†

Abstract

In this paper we study the preconditioning of n × n non-symmetric, real Toeplitz systems, when the generating function of the coefficient matrix Tn is not known a priori, but we know that a generating function f exists related to the matrix sequence {Tn}, Tn = Tn(f ), with f smooth enough. The proposed preconditioner is derived as a combination of a band Toeplitz and a circulant matrix. We give details for the construction of the proposed preconditioner, by the entries of Tn and we study the cluster of the eigenvalues, as well as of the singular values, of the sequences of the coefficient matrices related to the preconditioned systems. Theoretical results prove the efficiency of the Preconditioned Generalized Minimal Residual method (PGMRES) and the Preconditioned Conjugate Gradient method of Normal Equations (PCGN). Such efficiency is also shown in demonstrating numerical examples, using the proposed preconditioning technique. ©2022 Global-Science Press