Volume 2011, Issue 1 713795

Research Article

Open Access

Convergence of GAOR Iterative Method with Strictly α Diagonally Dominant Matrices

Guangbin Wang,

Corresponding Author

Guangbin Wang

[email protected]

Department of Mathematics, Qingdao University of Science and Technology, Qingdao 266061, China qust.edu.cn

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Hao Wen,

Hao Wen

Department of Mathematics, Qingdao University of Science and Technology, Qingdao 266061, China qust.edu.cn

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Ting Wang,

Ting Wang

Department of Mathematics, Qingdao University of Science and Technology, Qingdao 266061, China qust.edu.cn

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Guangbin Wang,

Corresponding Author

Guangbin Wang

[email protected]

Department of Mathematics, Qingdao University of Science and Technology, Qingdao 266061, China qust.edu.cn

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Hao Wen,

Hao Wen

Department of Mathematics, Qingdao University of Science and Technology, Qingdao 266061, China qust.edu.cn

Search for more papers by this author

Ting Wang,

Ting Wang

Department of Mathematics, Qingdao University of Science and Technology, Qingdao 266061, China qust.edu.cn

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First published: 15 November 2011

https://doi.org/10.1155/2011/713795

Citations: 1

Academic Editor: Yongkun Li

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Abstract

We discuss the convergence of GAOR method for linear systems with strictly α diagonally dominant matrices. Moreover, we show that our results are better than ones of Darvishi and Hessari (2006), Tian et al. (2008) by using three numerical examples.

1. Introduction

Sometimes we have to solve the following linear system:

()

where

()

is invertible. Yuan proposed a generalized SOR(GSOR) method to solve linear system (1.1) in [1]; afterwards, Yuan and Jin [2] established a generalized AOR(GAOR) method to solve linear system (1.1) as follows:

()

where

()

In [2–4], some people studied the convergence of GAOR method for solving linear systems Hy = f. In [2], Yuan and Jin studied the convergence of GAOR method and show that the GAOR method is better than the GSOR method under certain conditions. In [3], Darvishi and Hessari studied the convergence of GAOR method for diagonally dominant coefficient matrices and gave the regions of convergence. In [4], Tian et al. studied the convergence of GAOR method for strictly diagonally dominant coefficient matrices and gave the regions of convergence.

Sometimes, the coefficient matrices of linear systems Hy = f are not strictly diagonally dominant. In this paper, we will discuss the convergence of GAOR method for linear systems Hy = f with strictly α diagonally dominant matrices which need not to be strictly diagonally dominant.

Throughout this paper, we denote the set of all complex matrices by C^n×n, the spectral radius of iterative matrix L_ω,r by ρ(L_ω,r). And

()

Definition 1.1 (see [5].)Let A = (a_ij) ∈ C^n×n. If there exists α ∈ [0,1],

()

holds, then we call A as α diagonally dominant and denote it as A ∈ D₀(α). If all the inequalities are strict, then we denote it as A ∈ D(α).

Obviously, if A is a strictly diagonally dominant matrix (A ∈ SD), then A ∈ D(α). But not vice versa.

In [3], Darvishi and Hessari obtained the following result.

Theorem 1.2. Let H ∈ SD and assume that ω ≥ r ≥ 0. Then the sufficient condition for convergence of the GAOR method is

()

where J_i and K_i are the i-row sums of the modulus of the entries of J and K, respectively.

In [4], Tian et al. obtained the following result.

Theorem 1.3. If H ∈ SD, then the sufficient conditions for the convergence of the GAOR method are either

(i)
0 ≤ r ≤ 1 and 0 < ω ≤ 1 or
(ii)
|r | ≤ min _i{(1 − J_i)/K_i} and 0 < ω < 2/(1 + max _i(J + rK) _i).

This work is organized as follows. In Section 2, we obtain bound for the spectral radius of the iterative matrix L_ω,r of GAOR iterative method. In Section 3, we present two convergence theorems of GAOR method. In Section 4, we present three numerical examples to show that our results are better than ones of Theorems 1.2 and 1.3.

2. Upper Bound of the Spectral Radius of L_ω,r

In this section, we obtain upper bound of the spectral radius of iterative matrix L_ω,r.

Theorem 2.1. Let H ∈ D(α), then ρ(L_ω,r) satisfies the following inequality:

()

Proof. Let λ be an arbitrary eigenvalue of iterative matrix L_ω,r, then

()

Equation (2.2) holds if and only if

()

If (λ + ω − 1)I − ωJ − ωrK ∈ D(α), that is,

()

where (ωJ+ωrK)_ii denotes the diagonal element of matrix ωJ + ωrK. From [5], we know that

()

then λ is not an eigenvalue of iterative matrix L_ω,r, and when

()

λ is not an eigenvalue of iterative matrix L_ω,r.

When λ is an eigenvalue of iterative matrix L_ω,r, then there exists at least one i (i ∈ N), such that

()

that is,

()

Hence,

()

3. Convergence of GAOR Method

In this section, we investigate the convergence of GAOR method to solve linear system (1.1). We assume that H is a strictly α diagonally dominant coefficient matrix and get some sufficient conditions for the convergence of GAOR method.

Theorem 3.1. Let H ∈ D(α), then GAOR method is convergent if ω, r satisfy either

(I)
0 < ω ≤ 1 and
()

(II)
and
()

Proof. Since H ∈ D(α), then GAOR method is convergent if we have ρ(L_ω,r) < 1, that is,

()

Firstly, when 0 < ω ≤ 1, then

()

That is,

()

which leads to

()

Secondly, when 1 < ω, then

()

That is,

()

which leads to

()

then

()

Thus we complete the proof.

When r = 0, we can get the following theorem.

Theorem 3.2. Let H ∈ D(α), then ρ(L_ω,0) < 1 if

()

Proof. Because H ∈ D(α), from Theorem 1.2, when r = 0, we have

()

Firstly, when 0 < ω ≤ 1,

()

That is

()

Secondly, when 1 < ω < 2,

()

That is

()

which leads to

()

So ω must satisfy

()

Thus we complete the proof of Theorem 3.2.

4. Examples

In the following examples, we give the regions of convergence of GAOR method to show that our results are better than ones of Theorems 1.2 and 1.3.

Example 4.1. Let

()

Obviously, H is not a strictly diagonally dominant matrix, so we cannot use the results of Theorems 1.2 and 1.3. But H ∈ D(1/2), and

()

By Theorem 3.1, we obtain the following regions of convergence:

(I)
0 < ω ≤ 1 and |r| < 1/10

(II)
1 < ω < 24/23 and |r| < (2 − (23/12)ω)(6/5ω).

Example 4.2 (Example 1 of paper [4]). Let

()

It is easy to test that H ∈ D(1/4), and

()

By Theorem 3.1, we obtain the following regions of convergence:

(I)
0 < ω ≤ 1 and |r| < 3/4

(II)
1 < ω < 1.2 and |r| < ((18 − 5ω)/4ω).

In addition, H is a strictly diagonally dominant matrix.

By Theorem 1.3, we obtain the following regions of convergence:

(I)
0 ≤ r ≤ 1 and 0 < ω ≤ 1,
(II)
0 ≤ r < 0.3 and 0 < ω < 1.2,

(III)
−0.3 < r < 0 and 0 < ω < (18/(15 − 10r)).

By Theorem 1.2, we obtain the following region of convergence:

0 ≤ r ≤ ω and 0 < ω < (18/(15 + 10r)).

And we get Figure 1, where the regions of convergence got by Theorems 3.1 and 1.3, Theorem 1.2 are bounded by blue lines, red lines, green lines, respectively.

Description unavailable — **Figure 1**
Open in figure viewer PowerPoint

This figure shows that the regions of convergence got by Theorem 3.1 are larger than ones got by Theorems 1.2 and 1.3.

Example 4.3. Consider

()

where

()

Obviously, H is not a strictly diagonally dominant matrix, so we cannot use the results of Theorems 1.2 and 1.3. But H ∈ D(1/2), and

()

By Theorem 3.1, we obtain the following regions of convergence:

(I)
0 < ω ≤ 1 and |r| < (2208/3481)

(II)
1 < ω < (192/169) and |r| < ((18432 − 16224ω)/3481ω).

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).

References

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Convergence of GAOR Iterative Method with Strictly α Diagonally Dominant Matrices

Abstract

1. Introduction

2. Upper Bound of the Spectral Radius of L_ω,r

3. Convergence of GAOR Method

4. Examples

Acknowledgments

References

Citing Literature

Figures

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Convergence of GAOR Iterative Method with Strictly α Diagonally Dominant Matrices

Abstract

1. Introduction

2. Upper Bound of the Spectral Radius of Lω,r

3. Convergence of GAOR Method

4. Examples

Acknowledgments

References

Citing Literature

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2. Upper Bound of the Spectral Radius of L_ω,r