Research Article  Open Access
ZhaoNian Pu, XueZhong Wang, "Block Preconditioned SSOR Methods for Matrices Linear Systems", Journal of Applied Mathematics, vol. 2013, Article ID 213659, 7 pages, 2013. https://doi.org/10.1155/2013/213659
Block Preconditioned SSOR Methods for Matrices Linear Systems
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 GaussSeidel, 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 Hmatrix; if , , then is also an Hmatrix.
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 equalpartitioned 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.

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 HakKeung 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. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 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. View at: Publisher Site  MathSciNet
 W. Li and Z.y. You, “The multiparameters overrelaxation method,” Journal of Computational Mathematics, vol. 16, no. 4, pp. 367–374, 1998. View at: Google Scholar  Zentralblatt MATH  MathSciNet
 R. S. Varga, Matrix Iterative Analysis, PrenticeHall, Englewood Cliffs, NJ, USA, 1962. View at: MathSciNet
 L. Yu. Kolotilina, “Twosided bounds for the inverse of an Hmatrix,” Linear Algebra and Its Applications, vol. 225, pp. 117–123, 1995. View at: Publisher Site  Google Scholar  MathSciNet
 M. Neumann and R. J. Plemmons, “Convergence of parallel multisplitting iterative methods for Mmatrices,” Linear Algebra and Its Applications, vol. 8889, pp. 559–573, 1987. View at: Publisher Site  Google Scholar  MathSciNet
Copyright
Copyright © 2013 ZhaoNian Pu and XueZhong 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.