Research Article  Open Access
Some Results on Preconditioned MixedType Splitting Iterative Method
Abstract
We present a preconditioned mixedtype splitting iterative method for solving the linear system , where is a Zmatrix. And we give some comparison theorems to show that the rate of convergence of the preconditioned mixedtype splitting iterative method is faster than that of the mixedtype 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 mixedtype 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 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 mixedtype splitting iterative method with is faster than that of the preconditioned mixedtype splitting iterative method with or . Finally, we give one numerical example to illustrate our results.
2. Preconditioned MixedType Splitting Iterative Method
For the linear system (1), we consider its preconditioned form as follows: with the preconditioner ; that is,
We apply the mixedtype splitting iterative method to it and have the corresponding preconditioned mixedtype 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 Zmatrix. 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 Zmatrix. 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 Zmatrix 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 Zmatrix. 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 righthand side of the above inequality is more than zero. By Lemma 8, .(2)If , the righthand side of the above inequality is more than zero. By Lemma 8, .
Theorem 14. Let be a nonsingular Zmatrix. 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 righthand side of the above inequality is more than zero. By Lemma 8, .(2)If , the righthand 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 SteinRosenbergtype theorems like those in the papers [10–13]. The results (theorems and corollaries) in Section 3 are the comparisons of spectral radius of iterative matrices between the mixedtype splitting method and the preconditioned mixedtype splitting method, while the results in the papers [10–13] are the comparisons of spectral radius of iterative matrices between the parallel decompositiontype 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 10–14.
Table 1 shows that that the rate of convergence of the preconditioned mixedtype splitting method is faster than that of the mixedtype splitting method. And it shows that the rate of convergence of the preconditioned mixedtype splitting method with is faster than that of the preconditioned mixedtype splitting method with or .

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).
References
 G. H. Cheng, T. Z. Huang, and S. Q. Shen, “Note to the mixedtype splitting iterative method for Zmatrices linear systems,” Journal of Computational and Applied Mathematics, vol. 220, no. 12, pp. 1–7, 2008. View at: Publisher Site  Google Scholar
 D. M. Young, Iterative Solution of Large Linear Systems, Academic Press, New York, NY, USA, 1971.
 H. Schneider, “Theorems on Msplittings of a singular Mmatrix which depend on graph structure,” Linear Algebra and Its Applications C, vol. 58, pp. 407–424, 1984. View at: Google Scholar
 R. S. Varga, Matrix Iterative Analysis, PrenticeHall, Englewood Cliffs, NJ, USA, 1962.
 W. Li and W. Sun, “Modified GaussSeidel type methods and Jacobi type methods for Zmatrices,” Linear Algebra and Its Applications, vol. 317, no. 1–3, pp. 227–240, 2000. View at: Google Scholar
 A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, Academic Press, New York, NY, USA, 1979.
 A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, SIAM, Philadelphia, PA, USA, 1994.
 O. Axelsson, Iterative Solution Methods, Cambridge University Press, Cambridge, UK, 1994.
 T. Z. Huang, G. H. Cheng, and X. Y. Cheng, “Modified SORtype iterative method for Zmatrices,” Applied Mathematics and Computation, vol. 175, no. 1, pp. 258–268, 2006. View at: Publisher Site  Google Scholar
 Z.Z. Bai, “On the comparisons of the multisplitting unsymmetric aor methods for Mmatrices,” Calcolo, vol. 32, no. 34, pp. 207–220, 1995. View at: Publisher Site  Google Scholar
 Z.Z. Bai, “The generalized SteinRosenberg type theorem for the PDAORmethod,” Bulletin of the Institute of Mathematics Academia Sinica, vol. 19, pp. 329–335, 1997 (Chinese). View at: Google Scholar
 Z.Z. Bai, “A class of parallel decompositiontype relaxation methods for large sparse systems of linear equations,” Linear Algebra and Its Applications, vol. 282, no. 1–3, pp. 1–24, 1998. View at: Google Scholar
 Z.Z. Bai and R. Nabben, “Some properties of the block matrices in the parallel decompositiontype relaxation methods,” Applied Numerical Mathematics, vol. 29, no. 2, pp. 167–170, 1999. View at: Google Scholar
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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.