Relaxation parameters and composite refinement techniques

A composite refinement technique for two stationary iterative methods, one of them contains a relaxation parameter, is introduced. Four new techniques, Jacobi successive over relaxation (SOR) composite refinement (RJSOR), SOR Jacobi composite refinement (RSORJ), Gauss–Seidel (GS) SOR composite refinement (RGSSOR) and SOR with GS composite refinement (RSORGS) are compared with their classical forms. The efficient performance of the new forms is well established and confirmed through numerical example. The computational costs and the speed of convergence are considered. The decrease in the required number of iteration is established through the calculation of the spectral radius of the iteration matrices. It is illustrated that the convergence of Jacobi and Gauss–Seidel methods engage the divergence and extend the domain of convergence in the SOR method in the refinement technique. The calculations and graphs are performed by computer algebra system, Mathematica.