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Advances in Numerical Analysis
Volume 2013 (2013), Article ID 512084, 7 pages
http://dx.doi.org/10.1155/2013/512084
Research Article

Some Results on Preconditioned Mixed-Type Splitting Iterative Method

1Department of Mathematics, Qingdao University of Science and Technology, Qingdao 266061, China
2Department of Mathematics, Shanghai University, Shanghai 200444, China

Received 20 June 2013; Revised 14 September 2013; Accepted 27 September 2013

Academic Editor: Zhong-Zhi Bai

Copyright © 2013 Guangbin Wang and Fuping Tan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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 that . A matrix is an -matrix if , and , for all , such that .

Definition 2 (see [2]). A matrix is an -matrix if is a nonsingular -matrix, and .

Definition 3 (see [2, 3]). Let . Then, is called a regular splitting if and ; is called an -splitting if is an -matrix, .

Lemma 4 (see [2]). Let be an irreducible matrix. Then,(1) has a positive real eigenvalue equal to its spectral radius;(2)to , there corresponds an eigenvector ;(3) is a simple eigenvalue of .

Lemma 5 (see [4]). Let be a nonnegative matrix. Then,(1)if for some nonnegative vector , , then ;(2)if for some positive vector , then . Moreover, if is irreducible and if for some nonnegative vectors , then

Lemma 6 (see [5]). Let be an -splitting of . Then, if and only if is a nonsingular -matrix.

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 .

Lemma 8 (see [8]). Let be a regular splitting of . Then, the splitting is convergent if and only if .

Lemma 9 (see [9]). Let and be two nonsingular lower triangular -matrices. If , then .

3. Convergence Analysis and Comparison Results

Theorem 10. Let be a nonsingular Z-matrix. Assume that , , , and and are the iterative matrices given by (14) and (8), respectively. Consider the following.(i)If , then .(ii)Let be irreducible. Assume that and ;
then,one has(1), if ,(2), if .

Proof. Let
Then, we have (i) Since is a nonsingular Z-matrix and , , it is clear that is a nonsingular -matrix and the splitting is an -splitting. Since , it follows from Lemma 6 that is a nonsingular -matrix. Then, by Lemma 7, there is a positive vector such that , so .
By Lemma 7, 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 is a strictly lower triangular matrix, so that .
So, we have .
Then, ; hence, is a nonsingular -matrix.
For , it is obvious that . And for , we have . Thus, and .
We have proven that and are both -splittings and , two splittings are nonnegative.
On the other hand, since , we get which implies that
Therefore, . So, we have ; that is, (ii) Let be irreducible. Since is a nonnegative and irreducible matrix, and according to the proof of Lemma 4 in paper [9], we can obtain that and are nonnegative and irreducible matrices. Thus, from Lemma 4, we know that there exists a positive vector such that , where we denote , which is equivalent to Let , where , , and are the diagonal, lower triangular, and upper triangular parts of , respectively. So, where , , .
Now, we consider Since is an -matrix, and , we have the following.(1)If , then . By Lemma 5, we get .(2)If , then . By Lemma 5, we get .

Theorem 11. Let be a nonsingular -matrix. Assume that , , , and and are the iterative matrices given by (13) and (8), respectively. Consider the following.(i)If , then (ii)Let be irreducible. Assume that Then, one has (1) if ,(2) if .

Proof. Let
Then, we have
(i) By a similar proof of Theorem 10, we can prove that and are both -splitting and , two splittings , are nonnegative.
On the other hand, since , we get which implies that
Therefore, . So, we have ; that is,
(ii) Let where , , and , , , , , and are the diagonal, strictly lower, and strictly upper triangular matrices of and , respectively.
And denote ; then according to (35), we have
By (25), we have If , then by the proof of Theorem 10, we have .
Therefore, one has the following.(1)If , then but not equal to 0. Therefore, . By Lemma 5, we get .(2)If , then but not equal to 0. Therefore, . By Lemma 5, we get .
Remark. If we choose in Theorem 11, we have a similar result which is showed by the following corollary.

Corollary 12. Let be a nonsingular Z-matrix. Assume that , , , and and are the iterative matrices given by (15) and (8), respectively. Consider the following.(i)If , then (ii)Let be irreducible. Assume that
then, one has(1) if ,(2) if .
Now, one will provide some results to show the relations among , , and .

Theorem 13. Let be a nonsingular -matrix. Let and be iterative matrices given by (13) and (14), respectively. Assume that , , . If and , then(1) if ;(2) if .

Proof. Since and are two lower triangular -matrices with , by Lemma 9, we have
By the proof of Theorems 10 and 11, we consider
In view of the proof of Theorem 11, we have .
Therefore, one has the following.(1)If , the right-hand side of the above inequality is more than zero. By Lemma 8, .(2)If , the right-hand side of the above inequality is more than zero. By Lemma 8, .

Theorem 14. Let be a nonsingular Z-matrix. Let and be iterative matrices given by (13) and (15), respectively. Assume that , , . If and ,
then(1) if ,(2) if .

Proof. Since and are two lower triangular -matrices with , by Lemma 9, we have
By the proof of Corollary 12 and Theorem 11, we consider
Since , we get the following.(1)If , the right-hand side of the above inequality is more than zero. By Lemma 8, .(2)If , the right-hand side of the above inequality is more than zero. By Lemma 8, .
Remark. The results (theorems and corollaries) in Section 3 are in some sense the generalized Stein-Rosenberg-type theorems like those in the papers [1013]. 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 [1013] are the comparisons of spectral radius of iterative matrices between the parallel decomposition-type relaxation method and its special case.

4. Numerical Example

Consider the following equation: in the unit square with Dirichlet boundary conditions.

If we apply the central difference scheme on a uniform grid with interior nodes () 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 .

We choose ; then .

If we choose(1), , ,(2), , ,(3), , , then we can obtain the following results by Theorems 1014.

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 .

tab1
Table 1

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant no. 11001144) and the Natural Science Foundation of Shandong Province of China (ZR2012AL09).

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