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

Abstract

The SOR is a basic iterative method for solution of the linear system 𝐴𝑥=𝑏. Such systems can easily be solved using direct methods such as Gaussian elimination. However, when the coefficient matrix 𝐴 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 𝐴𝑥=𝑏 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 𝑀−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 𝐺1 and 𝐺2 are less than that of the classical SOR, which implies faster convergence.