Journal of Applied Mathematics

Volume 2013 (2013), Article ID 213659, 7 pages

http://dx.doi.org/10.1155/2013/213659

Research Article

Block Preconditioned SSOR Methods for -Matrices Linear Systems

Zhao-Nian Pu and Xue-Zhong Wang

School of Mathematics and Statistics, Hexi University, Zhangye, Gansu 734000, China

Received 9 January 2013; Accepted 12 March 2013

Academic Editor: Hak-Keung Lam

Copyright © 2013 Zhao-Nian Pu and Xue-Zhong Wang. 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 block preconditioner and consider block preconditioned SSOR iterative methods for solving linear system . When is an -matrix, the convergence and some comparison results of the spectral radius for our methods are given. Numerical examples are also given to illustrate that our methods are valid.

1. Introduction

For the linear system

where is an square matrix and and are -dimensional vectors. The basic iterative method for solving (1) is

where and is nonsingular. Thus (2) can be written as

where , .

Let us consider the following partition of :

where the blocks , , are nonsingular and .

Usually we split into

where , and are strictly block lower and strictly block upper triangular parts of , respectively. Let , and

Then, the iteration matrix of the SSOR method for is given by

Transforming the original system (1) into the preconditioned form

then we can define the basic iterative scheme:

where and is nonsingular. Thus (9) can also be written as

where ,. Similar to the original system (1), we call the basic iterative methods corresponding to the preconditioned system the preconditioned iterative methods.

When is an -matrix, Alanelli and Hadjidimosin [1] considered the preconditioner , where and is given by

with being the lower triangular matrix in the LU triangular decomposition of .

We consider the preconditioner , where

Let

where , and are block diagonally, strictly block lower, and strictly block upper triangular parts of , respectively. If is nonsingular, then and exist and it is possible to define the SSOR iteration matrix for . Namely,

Alanelli and Hadjidimos in [1] showed that the preconditioned Gauss-Seidel, the preconditioned SOR, and the preconditioned Jacobi methods with preconditioner are better than original methods. Our work in the presentation is to prove convergence of the block preconditioned SSOR method with preconditioner and give some comparison results of the spectral radius for the case when is an -matrix.

Let denote the matrix whose elements are the moduli of the elements of the given matrix. We call to comparison matrix if for , if for . For (4), under the previous definition, we have

Let , where , , and are block diagonally, strictly block lower, and strictly block upper triangular parts of , respectively.

Notice that the preconditioner of the matrix corresponding to is ; namely,

Let , where , , and are block diagonally, strictly block lower, and strictly block upper triangular parts of , respectively.

If is nonsingular, then and exist and the SSOR iteration matrix for is as follows:

2. Preliminaries

A matrix is called nonnegative (positive) if each entry of is nonnegative (positive). We denote it by (). Similarly, for -dimensional vector , we can also define (). Additionally, we denote the spectral radius of by . denotes the transpose of . A matrix is called a -matrix if for any , . A -matrix is a nonsingular -matrix if is nonsingular and , If is a nonsingular -matrix , then is called an -matrix. is said to be a splitting of if is nonsingular, is said to be regular if and , and weak regular if and , respectively.

Some basic properties on special matrices introduced previously are given to be used in this paper.

Lemma 1 (see [2]). Let A be a -matrix. Then the following statements are equivalent.

(a)is an -matrix.(b)There is a positive vector such that .(c).(d)All principal submatrices of are -matrices.(e)All principal minors are positive.

Lemma 2 (see [3, 4]). Let be an -matrix and let be a weak regular splitting. Then .

Lemma 3 (see [2]). Let and be two matrices with . Then .

Lemma 4 (see [5]). If is an -matrix, then .

Lemma 5 (see [6]). Suppose that and are weak regular splitting of monotone matrices and , respectively, such that . If there exists a positive vector such that , then for the monotone norm associated with ,

In particular, if has a positive Perron vector, then

Moreover if is a Perron vector of and strict inequality holds in (18), then strict inequality holds in (19).

Lemma 6. If and are two matrices, then .

Proof. It is easy to see that , for , and , for . Therefore, is true.

Lemma 7. If is an -matrix with unit diagonal elements, then .

Proof. Let , from being an -matrix; then and , and thus, we have

and then .

3. Convergence Results

Let , , , , , where and are partitioned in accordance with the block partitioning of the matrix , and let

Theorem 8. Let be a nonsingular H-matrix; if , , then is also an H-matrix.

Proof. From being an -matrix, we have , and . Let

Then

Therefore, is an -matrix, and then is an -matrix.

Theorem 9. If is a nonsingular -matrix with unit diagonal elements, and ,. Then .

Proof. From Theorem 8, we know is an -matrix; if we let

then the SSOR iteration matrix for is as follows:

Since is an -matrix; we have , and are -matrices; by simple calculation, we obtain that (24) is a weak regular splitting; from Lemma 2, we know that . Since

then, by Lemma 3, .

4. Comparison Results of Spectral Radius

Theorem 10. Let be a nonsingular -matrix with unit diagonal elements, and ,. Then is an -matrix and .

Proof. Similar to the proof of Theorems 8 and 9, it is easy to get the proof of this theorem.

In what follows we will give some comparison results on the spectral radius of preconditioned SSOR iteration matrices with different preconditioner.

Let

where

Then the SSOR iteration matrix for is as follows:

and let

where

Then the AOR iteration matrix for is (17).

Theorem 11. If is a nonsingular -matrix with unit diagonal elements, and ,. Then .

Proof. Since is a nonsingular -matrix, by Theorem 10, is a nonsingular -matrix, and thus and are two monotone matrices.

From and being -matrices, we can get , , , and are -matrices, together with

We obtain that and are two weak regular splittings. By simple calculation, we have

and thus ; letting , then ; since , we have

It follows that

As is a weak regular splitting, there exists a positive perron vector ; by Lemma 5, the following inequality holds:

that is,

When is a nonsingular -matrix, we have . If , , then . Furthermore, we have and ; therefore, we get the following result.

Corollary 12. Let be a nonsingular -matrix with unit diagonal elements, , , and . Then

Theorem 13. Let be a nonsingular -matrix with unit diagonal elements, and , . Then .

Proof . Let

Then the SSOR iteration matrix for is which is defined in the proof of Theorem 9, and let

Then the SSOR iteration matrix for is (17). It is easy to know that the previous two splittings are weak regular splittings. Furthermore, by Lemma 6, we have the following result, for any , ,

From and being two -matrices, we have

and then

Therefore, by Lemma 3, .

Combining the previous Theorems, we can obtain the following conclusion.

Theorem 14. Let be a nonsingular -matrix with unit diagonal elements, and , . Then

5. Numerical Example

For randomly generated nonsingular -matrices for with , we have determined the spectral radius of the iteration matrices of SSOR method mentioned previously with preconditioner . We report the spectral radius of the corresponding iteration matrix by . The parameters , , are taken from the equal-partitioned points of the interval . We take

For , we make two groups of experiments. In Figure 1, we test the relation between and , when , , where “×”, “+”, “”, “” and “” denote the spectral radius of , , , , and , respectively. In Table 1, the meaning of notations , , , , and denotes the spectral radius of , , , , and , respectively.

Table 1: Comparison of spectral radius with preconditioner .

)

100
0.5636
0.8635
0.7404
0.8999
0.6288

200
0.6030
0.9195
0.7698
0.9473
0.7059

500
0.6120
0.9751
0.7844
0.9847
0.7103

100
0.3650
0.7906
0.5923
0.8530
0.4770

200
0.4510
0.8832
0.6507
0.9240
0.5769

500
0.4345
0.9730
0.6766
0.9835
0.5609

100
0.2387
0.7512
0.6647
0.8087
0.3602

200
0.3494
0.8491
0.5800
0.9009
0.4899

500
0.3569
0.9561
0.6284
0.9731
0.5019

1000
0.3674
0.9738
0.6316
0.9840
0.5173

Figure 1: The relation between and , when , .

From Figure 1 and Table 1, we can conclude that the spectral radius of the preconditioned SSOR method with preconditioner is the best among others, which further illustrates that, Theorem 14 is true.

Acknowledgments

The authors express their thanks to the editor Professor Hak-Keung Lam and the anonymous referees who made much useful and detailed suggestions that helped them to correct some minor errors and improve the quality of the paper.

References

M. Alanelli and A. Hadjidimos, “Block Gauss elimination followed by a classical iterative method for the solution of linear systems,” Journal of Computational and Applied Mathematics, vol. 163, no. 2, pp. 381–400, 2004.

A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, vol. 9 of Classics in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, Pa, USA, 1994.

W. Li and Z.-y. You, “The multi-parameters overrelaxation method,” Journal of Computational Mathematics, vol. 16, no. 4, pp. 367–374, 1998.

R. S. Varga, Matrix Iterative Analysis, Prentice-Hall, Englewood Cliffs, NJ, USA, 1962.

L. Yu. Kolotilina, “Two-sided bounds for the inverse of an H-matrix,” Linear Algebra and Its Applications, vol. 225, pp. 117–123, 1995.

M. Neumann and R. J. Plemmons, “Convergence of parallel multisplitting iterative methods for M-matrices,” Linear Algebra and Its Applications, vol. 88-89, pp. 559–573, 1987.