REFINEMENT OF PRECONDITIONED OVERRELAXATION ALGORITHM FOR SOLUTION OF THE LINEAR ALGEBRAIC SYSTEM 𝑨𝒙=𝒃

Abstract

In this paper, a refinement of preconditioned successive overrelaxation method for solving the linear system 𝐵𝑥=𝑐 is considered. The coefficient matrix 𝐵∈𝑅𝑛,𝑛 is a nonsingular real matrix, 𝑐∈𝑅𝑛 and 𝑥 is the vector of unknowns. Based on the usual splitting of the coefficient matrix 𝐵 as 𝐵=𝐷−𝐿𝐵−𝑈𝐵, the linear system is expressed as 𝐴𝑥=𝑏 or (𝐼−𝐿−𝑈)𝑥=𝑏; where 𝐿=𝐷−1𝐿𝐵, 𝑈=𝐷−1𝑈𝐵 and 𝑏=𝐷−1𝑐. This system is further preconditioned with a preconditioner of the type 𝑃=𝐼+𝑆 as 𝐴̅𝑥=𝑏̅ or (𝐷̅−𝐿̅−𝑈̅)𝑥=𝑏̅. A refinement of the resulting preconditioned successive overrelaxation (SOR) method is performed. Convergence of the resulting refinement of preconditioned SOR iteration is established and numerical experiments undertaken to demonstrate the effectiveness and efficiency of the method. Results comparison revealed that the refinement of SOR method converges faster than the preconditioned as well as the classical SOR method.