M. Tzoumas

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.

On Iterative Solution for Linear Complementarity Problem with an

The numerous applications of the linear complementarity problem (LCP) in, e.g., the solution of linear and convex quadratic programming, free boundary value problems of fluid mechanics, and moving boundary value problems of economics make its efficient numerical solution a very imperative and interesting area of research. For the solution of the LCP, many iterative methods have been proposed, especially, when the matrix of the problem is a real positive definite or an

H_{+}

H_{+}

-Matrix

-matrix. In this work we assume that the real matrix of the LCP is an

On the Solution of the Linear Complementarity Problem by the Generalized Accelerated Overrelaxation Iterative Method

H_{+}

Towards the determination of the optimal p -cyclic SSOR

-matrix and solve it by using a new method, the scaled extrapolated block modulus algorithm, as well as an improved version of the very recently introduced modulus-based matrix splitting modified AOR iteration method. As is shown by numerical examples, the two new methods are very effective and competitive with each other. (A corrected PDF is attached to this article.)

ON THE EXACT P-CYCLIC SSOR CONVERGENCE DOMAINS

In the present work, we determine intervals of convergence for the various parameters involved for what is known as the generalized accelerated overrelaxation iterative method for the solution of the linear complementarity problem. The convergence intervals found constitute sufficient conditions for the generalized accelerated overrelaxation method to converge and are better than what have been known so far.

The solution of the linear complementarity problem by the matrix analogue of the accelerated overrelaxation iterative method

Abstract Suppose that A ∈ C n,n is a block p -cyclic consistently ordered matrix and let B and S ω denote the block Jacobi and the block symmetric successive overrelaxation (SSOR) iteration matrices associated with A , respectively. Extending previous work by Hadjidimos and Neumann, the present authors have determined the exact regions of convergence of the SSOR method in the ( ϱ ( B ), ω )-plane, for any p ⩾ 3, under the further assumption that the eigenvalues of B p are real of the same sign. In the present work the investigation goes on further, several questions are raised and among others the problem of the determination of the optimal regions of convergence in the spirit of Niethammer and Varga as well as that of the optimal relaxation factor are examined.

Exact SOR convergence regions for a general class ofp-cyclic matrices

Abstract Suppose that A ∈ Cn, n is a block p-cyclic consistently ordered matrix, and let B and Sω denote, respectively, the block Jacobi and the block symmetric successive overrelaxation (SSOR) iteration matrices associated with A. Neumaier and Varga found [in the (ϱ(|B|), ω) plane] the exact convergence and divergence domains of the SSOR method for the class of H-matrices. Hadjidimos and Neumann applied Rouches theorem to the functional equation connecting the eigenvalue spectra σ(B) and σ(Sω) obtained by Varga, Niethammer, and Cai, and derived in the (ϱ(B), ω) plane the convergence domains for the SSOR method associated with p-cyclic consistently ordered matrices, for any p ⩾ 3. In the present work it is further assumed that the eigenvalues of Bp are real of the same sign. Under this assumption the exact convergence domains in the (ϱ(B), ω) plane are derived in both the nonnegative and the nonpositive cases for any p ⩾ 3.

On sign symmetric circulant matrices

The Linear Complementarity Problem (LCP), with an H+−matrix coefficient, is solved by using the new “(Projected) Matrix Analogue of the AOR (MAAOR)” iterative method; this new method constitutes an extension of the “Generalized AOR (GAOR)” iterative method. In this work two sets of convergence intervals of the parameters involved are determined by the theories of “Perron-Frobenius” and of “Regular Splittings”. It is shown that the intervals in question are better than any similar convergence intervals found so far by similar iterative methods. A deeper analysis reveals that the “best” values of the parameters involved are those of the (projected) scalar Gauss-Seidel iterative method. A theoretical comparison of the “best” (projected) Gauss-Seidel and the “best” modulus-based splitting Gauss-Seidel method is in favor of the former method. A number of numerical examples support most of our theoretical findings.