Convergence of the preconditioned AOR method for irreducible L-matrices

Abstract

In this paper, we first point out some errors in a recent article by Li et al. [Y. Li, C. Li, S. Wu, Improving AOR method for consistent linear systems, Appl. Math. Comput. 186 (2007) 379–388], and then we provide some correct results for convergence of the preconditioned AOR method for irreducible L-matrices. Lastly, we provide numerical experiments to illustrate the theoretical results.

Introduction

In this paper, we consider the following linear system:

$$\mathit{Ax}=b\text{,}x\text{,}b\in {\mathbb{R}}^{\mathbb{n}}\text{,}$$

where $A=(a_{\mathit{ij}})\in {\mathbb{R}}^{\mathbb{n}\times \mathbb{n}}$ is a nonsingular matrix. The basic iterative method for solving the linear system (1) can be expressed as

$$x_{k+1}=M^{-1}{\mathit{Nx}}_k+M^{-1}b\text{,}k=0\text{,}1\text{,}\dots \text{,}$$

where $x_0$ is an initial vector and $A=M-N$ is a splitting of A. The $M^{-1}N$ is called an iteration matrix of the basic iterative method.

Throughout the paper, we assume that $A=I-L-U$, where I is the identity matrix, and L and U are strictly lower triangular and strictly upper triangular matrices, respectively. Then the iteration matrix of the AOR iterative method [3] for solving the linear system (1) is

$$T_{r\omega }=(I-\mathit{rL}{)}^{-1}((1-\omega )I+(\omega -r)L+\omega U)\text{,}$$

where $\omega$ and r are real parameters with $\omega \ne 0$.

In order to accelerate the convergence of iterative method for solving the linear system (1), the original linear system (1) is transformed into the following preconditioned linear system

$$\mathit{PAx}=\mathit{Pb}\text{,}$$

where P, called a preconditioner, is a nonsingular matrix. In this paper, we consider the following two cases where $P=P_1$ or $P=P_2$: The preconditioner $P_1$ introduced by Milaszewicz [5] is of the form $P_1=I+S_1$, where

$$S_1=\left(\begin{array}{cccc}\hfill 0\hfill & \hfill 0\hfill & \hfill \cdots \hfill & \hfill 0\hfill \\ \hfill -a_{21}\hfill & \hfill 0\hfill & \hfill \cdots \hfill & \hfill 0\hfill \\ \hfill -a_{31}\hfill & \hfill 0\hfill & \hfill \cdots \hfill & \hfill 0\hfill \\ \hfill ⋮\hfill & \hfill ⋮\hfill & \hfill ⋮\hfill & \hfill ⋮\hfill \\ \hfill -a_{n1}\hfill & \hfill 0\hfill & \hfill \cdots \hfill & \hfill 0\hfill \end{array}\right)\text{.}$$

The preconditioner $P_2$ introduced by Gunawardena et al. [2] is of the form $P_2=I+S_2$, where

$$S_2=\left(\begin{array}{ccccc}\hfill 0\hfill & \hfill -a_{12}\hfill & \hfill 0\hfill & \hfill \cdots \hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill -a_{23}\hfill & \hfill \cdots \hfill & \hfill 0\hfill \\ \hfill ⋮\hfill & \hfill ⋮\hfill & \hfill ⋮\hfill & \hfill ⋮\hfill & \hfill ⋮\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \cdots \hfill & \hfill -a_{n-1\text{,}n}\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \cdots \hfill & \hfill 0\hfill \end{array}\right)\text{.}$$

Let $\stackrel{\sim }{A}=P_{1}A$ and $S_{1}U=D^{\prime }+L^{\prime }+U^{\prime }$, where $D^{\prime }$ is a diagonal matrix, $L^{\prime }$ is a strictly lower triangular matrix, and $U^{\prime }$ is a strictly upper triangular matrix. Then, from $S_{1}L=0$ one obtains

$$\stackrel{\sim }{A}=(I+S_1)(I-L-U)=I-L-U+S_1-S_{1}U=\stackrel{\sim }{D}-\stackrel{\sim }{L}-\stackrel{\sim }{U}\text{,}$$

where $\stackrel{\sim }{D}=I-D^{\prime }$, $\stackrel{\sim }{L}=L-S_1+L^{\prime }$, and $\stackrel{\sim }{U}=U+U^{\prime }$.

Let $\overline{A}=P_{2}A$ and $S_{2}L=D^{\ast }+L^{\ast }$, where $D^{\ast }$ is a diagonal matrix and $L^{\ast }$ is a strictly lower triangular matrix. Then, one obtains

$$\overline{A}=(I+S_2)(I-L-U)=I-L-U+S_2-S_{2}L-S_{2}U=\overline{D}-\overline{L}-\overline{U}\text{,}$$

where $\overline{D}=I-D^{\ast }$, $\overline{L}=L+L^{\ast }$, and $\overline{U}=U-S_2+S_{2}U$.

If we apply the AOR iterative method to the preconditioned linear system (4), then we get the preconditioned AOR iterative method whose iteration matrix is

$${\stackrel{\sim }{T}}_{r\omega }=(\stackrel{\sim }{D}-r\stackrel{\sim }{L}{)}^{-1}((1-\omega )\stackrel{\sim }{D}+(\omega -r)\stackrel{\sim }{L}+\omega \stackrel{\sim }{U})\text{if}P=P_1\text{,}$$

or

$${\overline{T}}_{r\omega }=(\overline{D}-r\overline{L}{)}^{-1}((1-\omega )\overline{D}+(\omega -r)\overline{L}+\omega \overline{U})\text{if}P=P_2\text{.}$$

This paper is organized as follows. In Section 2, we present some notation, definitions and preliminary results. In Section 3, we first point out some errors in [4], and then we provide some correct results for convergence of the preconditioned AOR iterative method for irreducible L-matrices. In Section 4, we provide numerical experiments to illustrate the theoretical results obtained in Section 3.