Journal of Applied Mathematics

Volume 2012 (2012), Article ID 274636, 16 pages

doi:10.1155/2012/274636

Research Article

Improving Results on Convergence of AOR Method

Guangbin Wang, Ting Wang, and Yanli Du

Department of Mathematics, Qingdao University of Science and Technology, Qingdao 266061, China

Received 16 June 2012; Revised 27 August 2012; Accepted 4 October 2012

Academic Editor: Carlos J. S. Alves

Copyright © 2012 Guangbin Wang et al. 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 some sufficient conditions on convergence of AOR method for solving with being a strictly doubly diagonally dominant matrix. Moreover, we give two numerical examples to show the advantage of the new results.

1. Introduction

Let us denote all complex square matrices by and all complex vectors by .

For , we denote by the spectral radius of matrix .

Let us consider linear system , where is a given vector and is an unknown vector. Let be given and is the diagonal matrix, and are strictly lower and strictly upper triangular parts of , respectively, and denote

where .

Then the AOR method [1] can be written as

where

2. Preliminaries

We denote

For any matrix , the comparison matrix is defined by

Definition 2.1 (see [2]). A matrix is called a strictly diagonally dominant matrix if

A matrix is called a strictly doubly diagonally dominant matrix if

Definition 2.2 (see [3]). A matrix is called a strictly diagonally dominant matrix if there exits , such that

Definition 2.3 (see [4]). Let , if there exits such that

then is called a strictly doubly diagonally dominant matrix .

In [3, 5, 6], some people studied the convergence of AOR method for solving linear system and gave the areas of convergence. In [5], Cvetković and Herceg studied the convergence of AOR method for strictly diagonally dominant matrices. In [3], Huang and Wang studied the convergence of AOR method for strictly diagonally dominant matrices. In [6], Gao and Huang studied the convergence of AOR method for strictly doubly diagonally dominant matrices.

Theorem 2.4 (see [3]). Let , then AOR method converges for

Theorem 2.5 (see [6]). Let , then AOR method converges for

where

3. Upper Bound for Spectral Radius of

In the following, we present an upper bound for spectral radius of AOR iterative matrix for strictly doubly diagonally dominant coefficient matrix.

Lemma 3.1 (see [4]). If , then is a nonsingular H-matrix.

Theorem 3.2. Let , if , for  all  , then

where

Proof. Let be an eigenvalue of such that

that is,

If , then by Lemma 3.1, is nonsingular and is not an eigenvalue of iterative matrix , that is, if

then is not an eigenvalue of . Especially, if

then is not an eigenvalue of .

If is an eigenvalue of , then there exits at least a couple of , such that

that is,

Since , and the discriminant of the quadratic in satisfies , then the solution of (3.8) satisfies

So

4. Improving Results on Convergence of AOR Method

In this section, we present new results on convergence of AOR method.

Theorem 4.1. Let , then AOR method converges if satisfy either

where

Proof. It is easy to verify that for each , which satisfies one of the conditions (I)–(III), we have

Firstly, we consider case . Since be a diagonally dominant matrix, then by Lemma 3.1, we know that is a nonsingular H-matrix; therefore, is a nonsingular M-matrix, and it follows that from paper [7], holds for and for ,

If , then

by extrapolation theorem [6], we have .

If , then it remains to analyze the case

Since when ,

then . From

we have .

(1)When , it easy to verify that (4.8) holds. (2)When , since

then by and , , we have

It is easy to verify that the discriminant of the quadratic in satisfies , and so there holds

or

For , we have , it is in contradiction with ((4.8)b). Therefore, should be deleted.

Secondly, we prove .

(1)When ,  ,

By , we have

It is easy to verify that the discriminant of the quadratic in satisfies , and so there holds

By and , , we obtain

(2)When , ,

By , we have

It is easy to verify that the discriminant of the quadratic in satisfies , and so there holds

By and , , we obtain

Therefore, by (4.16) and (4.20), we get

Finally, we prove .

(1)When , ,

By , we have

It is easy to verify that the discriminant of the quadratic in satisfies , and so there holds

By , we obtain

(2)When ,  ,

By , we have

It is easy to verify that the discriminant of the quadratic in satisfies , and so there holds

By , we obtain

Therefore, by (4.25) and (4.29), we obtain

We can obtain the following results easily.

Theorem 4.2. Let . If , when , the following conditions hold:

or when , the following conditions hold:

then the area of convergence of AOR method obtained by Theorem 4.1 is larger than that obtained by Theorem 2.5.

Theorem 4.3. Let . If , when , the following conditions hold:

or when , the following conditions hold:

then the area of convergence of AOR method obtained by Theorem 4.1 is larger than that obtained by Theorem 2.5.

5. Examples

In the following examples, we give the areas of convergence of AOR method to show that our results are better than ones obtained by Theorems 2.4 and 2.5.

Example 5.1 (see [6]). Let

where

Obviously, , but .

By Theorem 4.1, we have the following area of convergence:

Obviously, .

By Theorem 2.5, we have the following area of convergence:

In addition, .

By Theorem 2.4, we have the following area of convergence:

Now we give two figures. In Figure 1, we can see that the area of convergence obtained by Theorem 4.1 (real line) is larger than that obtained by Theorem 2.5 (virtual line). In Figure 2, we can see that the area of convergence obtained by Theorem 4.1 (real line) is larger than that obtained by Theorem 2.4 (virtual line). From above we know that the area of convergence obtained by Theorem 4.1 is larger than ones obtained by Theorems 2.5 and 2.4.

Figure 1

Figure 2

Example 5.2. Let

Obviously, , , . So we cannot use Theorems 2.4 and 2.5. By Theorem 4.1, we have the following area of convergence:

Acknowledgments

The authors would like to thank the referees for their valuable comments and suggestions, which greatly improved the original version of this paper. This work was supported by the National Natural Science Foundation of China (Grant no. 11001144) and the Science and Technology Program of Shandong Universities of China (J10LA06).

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