Optimum extrapolated method in a special case with application to sor method

Abstract

Given a linear system Ax = b and an iterative method x ^{(m + 1)} = Gx ^(m)+k, m = 0,1,2,…(1) to solve it, we determine analytically the optimum extrapolation factor of the extrapolated method of (1), when all the eigenvalues of G have the same modulus (Section 2). Then using the SOR theory in the case of consistently ordered matrices A and applying the results of Section 2 to the extrapolated SOR (ESOR) method, we show (Section 3), that the globally optimum parameters of it (and also of the AOR method) [2, 11, 14] are recovered.

Keywords:

• Extrapolated method
• successive overrelaxation (SOR) method
• accelerated overrelaxation (AOR) method
• consistently ordered matrices