Toeplitz matrices: spectral properties and preconditioning in the CG method

We consider multilevel Toeplitz matrices Tn(f) generated by Lebesgue integrable functionsf defined over I, I = [−π, π), d ≥ 1. We are interested in the solution of linear systems withcoefficient matrix Tn(f) when the size of Tn(f) is large. Therefore the use of iterative methodsis recommended for computational and numerical stability reasons. In this note we focus ourattention on the (preconditioned) conjugate gradient (P)CG method and on the case where thesymbol f is known and univariate (d = 1): the second section treat spectral properties of Toeplitzmatrices Tn(f); the third deals with the spectral behavior of T−1n (g)Tn(f) and the fourth with theband Toeplitz preconditioning; in the fifth section we consider the matrix algebra preconditioningthrough the Korovkin theory. Then in the sixth section we study the multilevel case d > 1 byemphasizing the results that have a plain generalization (those in the Sections 2, 3, and 4) andthe results which strongly depend on the number d of levels (those in Section 5): in particularthe quality of the matrix algebra preconditioners (circulants, trigonometric algebras, Hartley etc.)deteriorates sensibly as d increases.A section of conclusive remarks and two appendices treating the theory of the (P)CG methodand spectral distributional results of structured matrix sequences.key words Linear system, conjugate gradient method, Toeplitz matrix, structured matrix, precon-ditioner.