On the convergence of a class of parallel decomposition-type relaxation methods (Q1354246)

The paper describes the development of a new class of parallel decomposition-type accelerated overrelaxation (AOR) methods, which are suitable to the SIMD-systems. Thereby, a general block form of a linear system of equations with a nonsingular matrix is considered. At first, the parallel decomposition-type Jacobi method, the parallel decomposition-type Gauss-Seidel method, and the parallel decomposition-type successive overrelaxation (SOR) method are generated by fixing the values of the two relaxation parameters that can be arbitrarily chosen in the parallel decomposition-type AOR method. Furthermore, sufficient and (or) necessary conditions for the convergence of the various schemes are presented for the cases that the coefficient matrix is an \(L\)-matrix, \(H\)-matrix, and a symmetric positive definite matrix, respectively. Finally, numerical results concerning two concrete examples are furnished to demonstrate the efficiency of the method as well as the validity of the convergence theories.