Spectral Properties of Preconditioned Rational Toeplitz Matrices: The Nonsymmetric Case

Inverse product Toeplitz preconditioners for non-Hermitian Toeplitz systems

“…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.…”

“…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.…”

“…The solution of non-Hermitian Toeplitz systems has been investigated by a number of researchers [4,8,9,11,12,17,20,22,23]. Circulant preconditioners and skew-circulant preconditioners [8,9], trigonometric preconditioners [12,22], Toeplitz-circulant preconditioners [4,11] and banded Toeplitz preconditioners [17] for (1.1) have been proposed.…”

Preconditioners for non-Hermitian Toeplitz systems

“…Although there exists a rich literature on Hermitian Toeplitz systems (see Reference [1] and the references therein), only few papers consider the non-Hermitian case [2][3][4][5][6]. These methods require in each iteration step only multiplications of A N (f) with vectors and since A N (f) is Toeplitz these multiplications can be computed in O(N log N ) arithmetical operations by using fast Fourier transforms.…”

“…Paper [5] contains interesting results for Strang preconditioners and for four other special preconditioners for the case that the generating function f¿0 is a rational function. The only paper, where generating functions with zeros were considered, is Reference [2].…”