On the convergence of a class of parallel decomposition-type relaxation methods
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.
scientific article; zbMATH DE number 1006539