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.


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 = −1 and = −1 . 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 +1 = + , = 0, 1, . . . , where = −1 and = −1 . In paper [1], Cheng et al. presented the mixed-type splitting iterative method as follows: with the following iterative matrix: where 1 is an auxiliary nonnegative diagonal matrix, 1 is an auxiliary strictly lower triangular matrix, and 0 ≤ 1 ≤ . In this paper, we will establish the preconditioned mixedtype 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.

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 , 1 is an auxiliary nonnegative diagonal matrix, 1 is an auxiliary strictly lower triangular matrix, and 0 ≤ 1 ≤ . If we choose = 0, we have the following corresponding iterative matrix: And if we choose = 0, 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.
Lemma 7 (see [6,7]). Let be a Z-matrix. Then, is a nonsingular -matrix if and only if there is a positive vector such that ≥ 0.

Convergence Analysis and Comparison Results
Theorem 10. Let be a nonsingular Z-matrix. Assume that matrices given by (14) and (8), respectively. Consider the following.
then,one has Proof. Let Then, we have (i) Since is a nonsingular Z-matrix and 1 ≥ 0, 0 ≤ 1 ≤ , it is clear that = + 1 + 1 − is a nonsingularmatrix and the splitting is an -splitting. Since ( ) < 1, it follows from Lemma 6 that is a nonsingular -matrix. Then, by Lemma 7, there is a positive vector such that is also a nonsingular -matrix. Obviously, we can get that is a positive diagonal matrix. And from is nonnegative, we know that being a -matrix. Since −1 ≥ 0 is a strictly lower triangular matrix, so that ( −1 ) = 0 < 1.
Therefore, one has the following.
Remark. If we choose = 0 in Theorem 11, we have a similar result which is showed by the following corollary. 1], and̂and are the iterative matrices given by (15) and (8), respectively. Consider the following.

Corollary 12. Let be a nonsingular Z-matrix. Assume that
(ii) Let be irreducible. Assume that then, one has Now, one will provide some results to show the relations among (̃), ( ), and (̂).
Therefore, one has the following.
Remark. The results (theorems and corollaries) in Section 3 are in some sense the generalized Stein-Rosenberg-type theorems like those in the papers [10][11][12][13]. The results (theorems and corollaries) in Section 3 are the comparisons of spectral radius of iterative matrices between the mixed-type splitting method and the preconditioned mixed-type splitting method, while the results in the papers [10][11][12][13] are the comparisons of spectral radius of iterative matrices between the parallel decomposition-type relaxation method and its special case.

Numerical Example
Consider the following equation: in the unit square Ω with Dirichlet boundary conditions. If we apply the central difference scheme on a uniform grid with × interior nodes ( 2 = ) to the discretization of the above equation, we can get a system of linear equations with the coefficient matrix where ⊗ denotes the Kronecker product, are × tridiagonal matrices, and the step size is ℎ = 1/ .
Advances in Numerical Analysis 7   Table 1 shows that that the rate of convergence of the preconditioned mixed-type splitting method is faster than that of the mixed-type splitting method. And it shows that the rate of convergence of the preconditioned mixed-type splitting method with is faster than that of the preconditioned mixed-type splitting method with or .