Nested Iterative Methods For Large And Sparse Linear Systems

Nested iterative method and nested block iterative methd for large sparse linear systems were presented by Lanzkron. etc.[24]. The convergence and the monotonicity for these two methods were studied when the coefficient matrix was monotone in [24]. Cao[8] [9] then further studied these two methods. The coefficient matrix in [9] was Hermitian matrix and the concept of prior splitting was developed in [8],In this paper, the two metheods in [24] are firther studied. We first discuss the convergence for these two methods when the coeificient matrix is H-matrix and the monotonicity for the nested iterative method when the coefficient matrix is monotone. Then, we introduce a new papallel iterative method which we call nested multisplitting methed by using the multisplitting method[32]. Convergence for this new method is studied when the coefficient matrices are either monotone matrices or H-matrices. Numerical tests are carried out in the end to validate our results. The numerical results also show that the nested iterative method has the same as or even better numerical performance than the splitting method in some aspects under the conditions in our tests.