Modified successive overrelaxation (SOR) type methods for M-matrices

The SOR is a basic iterative method for solution of the linear system =. Such systems can easily be solved using direct methods such as Gaussian elimination. However, when the coefficient matrix is large and sparse, iterative methods such as the SOR become indispensable. A new preconditioner for speeding up the convergence of the SOR iterative method for solving the linear system = is proposed. Arising from the preconditioner, two new preconditioned iterative techniques of the SOR method are developed. The preconditioned iterations are applied to the linear system whose coefficient matrix is an −matrix. Convergence of the preconditioned iterations is established through standard procedures. Numerical examples and results comparison are in conformity with the analytic results. More so, it is established that the spectral radii of the proposed preconditioned SOR 1 and 2 are less than that of the classical SOR, which implies faster convergence

The solution of linear systems of equations with a structural analysis code on the NAS CRAY-2

Two methods for solving linear systems of equations on the NAS Cray-2 are described. One is a direct method; the other is an iterative method. Both methods exploit the architecture of the Cray-2, particularly the vectorization, and are aimed at structural analysis applications. To demonstrate and evaluate the methods, they were installed in a finite element structural analysis code denoted the Computational Structural Mechanics (CSM) Testbed. A description of the techniques used to integrate the two solvers into the Testbed is given. Storage schemes, memory requirements, operation counts, and reformatting procedures are discussed. Finally, results from the new methods are compared with results from the initial Testbed sparse Choleski equation solver for three structural analysis problems. The new direct solvers described achieve the highest computational rates of the methods compared. The new iterative methods are not able to achieve as high computation rates as the vectorized direct solvers but are best for well conditioned problems which require fewer iterations to converge to the solution

Implicit ODE solvers with good local error control for the transient analysis of Markov models

Obtaining the transient probability distribution vector of a continuous-time Markov chain (CTMC) using an implicit ordinary differential equation (ODE) solver tends to be advantageous in terms of run-time computational cost when the product of the maximum output rate of the CTMC and the largest time of interest is large. In this paper, we show that when applied to the transient analysis of CTMCs, many implicit ODE solvers are such that the linear systems involved in their steps can be solved by using iterative methods with strict control of the 1-norm of the error. This allows the development of implementations of those ODE solvers for the transient analysis of CTMCs that can be more efficient and more accurate than more standard implementations.Peer ReviewedPostprint (published version