Convergence analysis of the two-stage¶multisplitting method

An example is given which shows that the asymptotic convergence rate of the two-stage multisplitting method (see D.B. Szyld and M.T. Jones, SIAM J. Matrix Anal. Appl. 13, 671–679 (1992)) with one inner iteration is, generally, either faster or slower than that with many inner iterations. When the coefficient matrix is an H-matrix and a monotone matrix, respectively, we formulate the convergence as well as the monotone convergence theories for this two-stage multisplitting method under suitable constraints on the two-stage multisplitting. Furthermore, the corresponding comparison theorem in the sense of monotonicity for this method is established and several concrete applications are discussed.