Some Results on Preconditioned Mixed-Type Splitting Iterative Method

We present a preconditioned mixed-type splitting iterative method for solving the linear system , where is a Z-matrix. And we give some comparison theorems to show that the rate of convergence of the preconditioned mixed-type splitting iterative method is faster than that of the mixed-type splitting iterative method. Finally, we give one numerical example to illustrate our results. 1. Introduction For solving linear system, where is an square matrix and and are -dimensional vectors, the basic iterative method is where and is nonsingular. Thus, (2) can be written as where and . Assuming that has unit diagonal entries, let , where is the identity matrix and and are strictly lower and strictly upper triangular parts of , respectively. Transform the original system (1) into the preconditioned form as follows: Then, we can define the basic iterative scheme as follows: where and is nonsingular. Thus, the equation above can also be written as where and . In paper [1], Cheng et al. presented the mixed-type splitting iterative method as follows: with the following iterative matrix: where is an auxiliary nonnegative diagonal matrix, is an auxiliary strictly lower triangular matrix, and . In this paper, we will establish the preconditioned mixed-type splitting iterative method with the preconditioners , , and for solving linear systems. And we obtain some comparison results which show that the rate of convergence of the preconditioned mixed-type splitting iterative method with is faster than that of the preconditioned mixed-type splitting iterative method with or . Finally, we give one numerical example to illustrate our results. 2. Preconditioned Mixed-Type Splitting Iterative Method For the linear system (1), we consider its preconditioned form as follows: with the preconditioner ; that is, We apply the mixed-type splitting iterative method to it and have the corresponding preconditioned mixed-type splitting iterative method as follows: that is, So, the iterative matrix is where , , and are the diagonal, strictly lower, and strictly upper triangular matrices obtained from , is an auxiliary nonnegative diagonal matrix, is an auxiliary strictly lower triangular matrix, and . If we choose , we have the following corresponding iterative matrix: And if we choose , we have the following corresponding iterative matrix: If we choose certain auxiliary matrices, we can get the classical iterative methods as follows.(1)The PSOR method is (2)The PAOR method is We need the following definitions and results. Definition 1 (see [2]). A matrix is a -matrix if , for all , such