Convergence of Preconditioned Gauss-Seidel Iterative Method For 𝑳 −Matrices

Abstract

A great many real-life situations are often modeled as linear system of equations, 𝐴𝑥 =𝑏. Direct methods of solution of such systems are not always realistic, especially where the coefficient matrix 𝐴 is very large and sparse, hence the recourse to iterative solution methods. The Gauss-Seidel, a basic iterative method for linear systems, is one such method. Although convergence is rarely guaranteed for all cases, it is established that the method converges for some situations depending on properties of the entries of the coefficient matrix and, by implication, on the algebraic structure of the method. However, as with all basic iterative methods, when it does converge, convergence could be slow. In this research, a preconditioned version of the Gauss- Seidel method is proposed in order to improve upon its convergence and robustness. For this purpose, convergence theorems are advanced and established. Numerical experiments are undertaken to validate results of the proved theorems.