D. Noutsos

More on modifications and improvements of classical iterative schemes for M-matrices

In the last four decades many articles have been devoted to the modifications and improvements of classes of preconditioners for linear systems whose matrix coefficient is an M-matrix in order to improve on the convergence rates of the classical iterative schemes (Jacobi, Gauss–Seidel, etc.). The present work is a contribution towards the generalization of the most common preconditioners used so far.

Matrix algebra preconditioners for multilevel Toeplitz systems do not insure optimal convergence rate

In the last decades several matrix algebra optimal and superlinear preconditioners (those assuring a strong clustering at the unity) have been proposed for the solution of polynomially ill-conditioned Toeplitz linear systems. The corresponding generalizations for multilevel structures are neither optimal nor superlinear (see e.g. Contemp. Math. 281 (2001) 193). Concerning the notion of superlinearity, it has been recently shown that the proper clustering cannot be obtained in general (see Linear Algebra Appl. 343-344 (2002) 303; SIAM J. Matrix Anal. Appl. 22(1) (1999) 431; Math. Comput. 72 (2003) 1305). In this paper, by exploiting a proof technique previously proposed by the authors (see Contemp. Math. 323 (2003) 313), we prove that the spectral equivalence and the essential spectral equivalence (up to a constant number of diverging eigenvalues) are impossible too. In conclusion, optimal matrix algebra preconditioners in the multilevel setting simply do not exist in general and therefore the search for optimal iterative solvers should be oriented to different directions with special attention to multilevel/multigrid techniques.

On the equivalence to the k -step iterative Euler methods and successive overrelaxation (SOR) methods for k -cyclic matrices

For the solution of the linear system x = Tx + c (1), where T is weakly cyclic of index k ≥ 2, the block SOR method together with two classes of monoparametric k-step iterative Euler methods, whose (optimum) convergence properties were studied in earlier papers, are considered. By establishing the existence of the matrix analog of the Vargas relation, connecting the eigenvalues of the SOR and the Jacobi matrices associated with (1), it is proved that the aforementioned SOR method is equivalent to a certain monoparametric k-step iterative Euler method derived from (1). By suitably modifying the existing theory, one can then determine (optimum) relaxation factors for which the SOR method in question converges, (optimum) regions of convergence etc., so that one can obtain, what is known, several new results. Finally, a number of theoretical applications of practical importance is also presented.

On the convergence of monoparametric k-step iterative euler methods for the solution of linear systems

For the solution of the nonsingular linear system x = Tx+c, two monoparametric stationary k—step iterative methods are considered. By using Euler transforms and for various values of their parameter ω the two methods are analyzed and studied as regards: (i) Their (optimum) convergence for a given configuration of the spectrum σ(T) of T and (ii) Their region of convergence R k , for a permissible ω, for all Ts for which σ(T) ⊂ R k . Answers to both questions are given and it is shown that if the two methods share the same quantity ρ, defined in the paper, the optimum second method is asymptotically much faster than the optimum first one.

On different classes of monoparametric stationary iterative methods for the solution of linear systems

The class of monoparametric k-step methods x(m)=ωTx(m−1)+(1−ω)x(m−k)+ωc used for the solution of the linear system (I − T)x = c is studied. Under certain conditions the spectrum σ(T) of T must satisfy, for ω >1 and given k(⩾ 2) and p∈ (0, 1) (a quantity defined in the paper), (optimum) convergent methods (1) are derived. Next, an equivalence between (optimum) convergent methods (1) and a class of Successive Overrelation (SOR.) ones is established. Then, based on (1), a class of new monoparametric methods, called k2-step block iterative methods faster than those in (1), is introduced and studied. Finally, various applications and numerical examples in support of the theory developed in this paper are provided.

Two-level Toeplitz preconditioning: approximation results for matrices and functions

Large

Block band Toeplitz preconditioners derived from generating function approximations: analysis and applications

2

Optimal stretched parameters for the SOR iterative method

-level Toeplitz systems arise in a variety of applications (see, e.g., [R. H. Chan and M. Ng, SIAM Rev., 38 (1996), pp. 427-482]) for which efficient numerical methods for their solution are required. Some successful numerical techniques need the explicit knowledge of the generating function

A preconditioning proposal for ill‐conditioned Hermitian two‐level Toeplitz systems

f

Towards the determination of the optimal p -cyclic SSOR

of the considered system