GRPM-style methods and comparisons of convergent and divergent rates

Relaxed technique is one of techniques for improving convergence rate of splitting iterative methods. Based on local relaxed method and system relaxed method of parallel multisplitting Frommer and Mayer [A. Frommer, G. Mayer, Convergence of relaxed parallel multisplitting methods, Linear Algebra Appl. 119 (1989) 141–152], we give the global relaxed parallel multisplitting (GRPM) method by introducing some relaxed parameters and study the convergence of our methods (GRPM-style) when the coefficient matrices are HH-matrices. Numerical experiments show that, when choosing the approximately optimal relaxed parameters, our methods have faster convergent rate than the methods in Chang [D.W. Chang, Convergence analysis of the parallel multisplitting TOR methods, J. Comput. Appl. Math. 72 (1996) 169–177] Frommer and Mayer [A. Frommer, G. Mayer, Convergence of relaxed parallel multisplitting methods, Linear Algebra Appl. 119 (1989) 141–152]. Furthermore, the convergent and divergent rates of local relaxed parallel multisplitting (LRPM-style) methods about multislitting TOR, AOR, SOR, G-S, extraolated Jacobi methods as well as Jacobi iterative method are compared in detail.