Convergence theory for the general GAOR type iterative method and the MSOR iterative method applied to H-matrices

Abstract

We discuss the general GAOR type (GGAOR) iterative method, of which special cases are the AOR, the GAOR, and the MSOR iterative methods, to solve the linear system Ax = b where A = I − L − U with L and U being general matrices. The GGAOR iteration matrix is expressed by L RΩ = (I − RL) −1[(I − Ω) + (Ω − R)L + ΩU] where R and Ω are diagonal matrices, and the MSOR iteration matrix L ωω′ = (I − Ω) −1(I − Ω + ΩU) . Some basic results on the upper bounds of the spectral radii ϱ(L RΩ) and ϱ( L ωω′ ) are given when A is strictly or irreducibly diagonally dominant by rows. Based upon these results, we obtain new results on the convergence regions of the GGAOR and the MSOR iterative methods when A is an H-matrix or A has property A, and recover and improve previous ones.