We introduce a multigrid technique for the solution of multilevel circulant linear systems whose coefficient matrix has eigenvalues of the form f (x [n] j ), where f is continuous and independent of n = (n 1 , . . . , n d ), and xThe interest of the proposed technique pertains to the multilevel banded case, where the total cost is optimal, i.e., O(N ) arithmetic operations (ops), N = d r=1 nr, instead of O(N log N ) ops arising from the use of FFTs. In fact, multilevel banded circulants are used as preconditioners for elliptic and parabolic PDEs (with Dirichlet or periodic boundary conditions) and for some two-dimensional image restoration problems where the point spread function (PSF) is numerically banded, so that the overall cost is reduced from O(k(ε, n) , n) is the number of PCG iterations to reach the solution within an accuracy of ε. Several numerical experiments concerning one-rank regularized circulant discretization of elliptic 2q-differential operators over one-dimensional and two-dimensional square domains with mixed boundary conditions are performed and discussed.

Summary.We study the role of preconditioning strategies recently developed for coercive problems in connection with a two-step iterative method, based on the Hermitian skew-Hermitian splitting (HSS) of the coefficient matrix, proposed by Bai, Golub and Ng for the solution of nonsymmetric linear systems whose real part is coercive. As a model problem we consider Finite Differences (FD) matrix sequences {A n (a, p)} n discretizing the elliptic (convection-diffusion) problem  

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In this paper, we propose and analyse preconditioning strategies for non-Hermitian Toeplitz linear systems. These techniques used in connection with the GMRES algorithm allow to solve in an optimal way the corresponding linear systems