W. S. Yousif

EXPLICIT DE-COUPLED GROUP ITERATIVE METHODS AND THEIR PARALLEL IMPLEMENTATIONS

In this paper we extend the 4-point explicit de-coupled group (EDG) iterative method, Abdullah (1991), to the 6 and 9-poinl EDG methods for the solution of elliptic partial differential equations. We will show graphically the technique of implementing the new grouping. Performance results for the algorithms are presented and a comparison with the 4-point scheme confirm the new groups to be computationally superior. Further, the implementations of the parallel 4, 6 and 9-point EDG methods on the Sequent Balance 8000 multiprocessor arc discussed and results from experiments performed are presented.

The implementation of the explicit block iterative methods on the balance 8000 parallel computer

Abstract The explicit block iterative method for solving elliptic p.d.e.s was introduced by Evans and Biggins [1] whilst in Yousif and Evans [2], larger size block methods were studied and their advantages investigated and compared with other iterative methods. In this paper, several variants of the implementation of these block methods on the Balance 8000 parallel computer are discussed.

A variant of the AOR method for augmented systems

In this paper we present a variant of the Accelerated Overrelaxation iterative method (AOR), denoted by modified AOR-like method (MAOR-like method) for solving the augmented systems, i.e. the AOR-like method with three real parameters , r and . For special values of r ,  and  we get the MSOR-like method, the AOR-like method and the SOR-like method. An equation relating the involved parameters and the eigenvalues of the iteration matrix of the MAOR-like method is obtained. Furthermore, some convergence conditions for the MAOR-like method are derived. This paper generalizes the main results of Li, Li, Nie, and Evans 2004 and Shao, Li, and Li (2007). Numerical examples are presented to show that, for a suitable choice of the involved parameters, the MAOR-like method is superior when compared to the above iterative methods and to the SSOR-like method presented by Zheng, Wang, and Wu (2009).

Further results on the preconditioned sor method

Several methods have been developed on the preconditioned iterative methodsi. e iterative methods applied to preconditioned linear systems. Usui, Kohno and Niki [4] have proposed the adaptive Gauss-Seidel (GS) method, and the same authors [5] have presented the pre-conditioned SOR method. They have shown, with the aid of numerical examples, that the two methods have a better rate of convergence in comparison with the classical SOR method. In this paper we will prove theoretically the improvement in the rate of convergence.

Explicit solution of block tridiagonal systems of linear equations

Abstract An explicit block tridiagonal solver suitable for parallel implementation is described for the equations derived from finite element discretisation of a two-point boundary value problem.

THE QZ ALGORITHM FOR THE CALCULATION OF THE EIGENVALUES OF A REAL MATRIX

Abstract In this paper we present a new algorithm for calculating the eigenvalues of a real matrix. The algorithm is based on the orthogonal decomposition of a square dense matrix by the QZ method proposed in [1]. A comparison with the QR algorithm confirm the new algorithm to be computationally superior.

The solution of unsymmetric tridiagonal Toeplitz systems by the strides reduction algorithm

Abstract A cyclic reduction method is described for the fast numerical solution of constant tridiagonal Toeplitz linear systems which occur repeatedly in the solution of the implicit finite difference equations derived from linear first order hyperbolic equations, i.e. the Transport equation, under a variety of boundary conditions. In this paper, we show that the linear systems can be solved efficiently by the Stride of 3 reduction algorithm.

SOLVING TRIDIAGONAL LINEAR SYSTEMS BY THE ENHANCED PARALLEL STRIDE OF THREE REDUCTION METHOD

In this paper, the parallelisation of the stride of three method for the solution of a tridiagonal system of equations for P processors is investigated. The presented algorithm is organised in such a way that all processors are fully operational at every stage of the solution process. The results of experiments carried out on the Sequent Balance 8000 multiprocessor are presented.

The accelerated overrelaxation quadrant interlocking iterative method

In this paper we use a new splitting of the matrix A of the linear system A x = b introduced in [1] and we present a new version of the AOR method, more suitable for parallel processing, which involves explicit evaluation of 2 × 2 blocks. We also obtain several convergence conditions for this new method, when the matrix A of (1.1) belongs to different classes of matrices. Some results, given in [1], are also improved and generalised.