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Journal of Applied Mathematics
Volume 2012, Article ID 563586, 9 pages
http://dx.doi.org/10.1155/2012/563586
Research Article

Comparison Results on Preconditioned GAOR Methods for Weighted Linear Least Squares Problems

1Department of Mathematics, Qingdao University of Science and Technology, Qingdao 266061, China
2Department of Mathematics, Shanghai University, Shanghai 200444, China

Received 18 July 2012; Accepted 26 August 2012

Academic Editor: Zhongxiao Jia

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 preconditioned generalized accelerated overrelaxation methods for solving weighted linear least square problems. We compare the spectral radii of the iteration matrices of the preconditioned and the original methods. The comparison results show that the preconditioned GAOR methods converge faster than the GAOR method whenever the GAOR method is convergent. Finally, we give a numerical example to confirm our theoretical results.

1. Introduction

Consider the weighted linear least squares problem where is the variance-covariance matrix. The problem has many scientific applications. A typical source is parameter estimation in mathematical modeling.

This problem has been discussed in many books and articles. In order to solve it, one has to solve a nonsingular linear system as where is an invertible matrix with

Yuan proposed a generalized SOR (GSOR) method to solve linear system (1) in [1]; afterwards, Yuan and Jin [2] established a generalized AOR (GAOR) method to solve linear system (1). In [3, 4], authors studied the convergence of the GAOR method for solving the linear system . In [3], authors studied the convergence of the GAOR method when the coefficient matrices are consistently ordered matrices and gave the regions of convergence. In [4], authors studied the convergence of the GAOR method for diagonally dominant coefficient matrices and gave the regions of convergence.

In order to solve the linear system (1.2) using the GAOR method, we split as

Then, for , one GAOR method can be defined by where is the iteration matrix and

In order to decrease the spectral radius of , an effective method is to precondition the linear system (1.2), namely, then the preconditioned GAOR method can be defined by where

In [5], authors presented three kinds of preconditioners for preconditioned modified accelerated overrelaxation method to solve systems of linear equations. They showed that the convergence rate of the preconditioned modified accelerated overrelaxation method is better than that of the original method, whenever the original method is convergent.

This paper is organized as follows. In Section 2, we give some important definition and the known results as the preliminaries of the paper. In Section 3, we propose three preconditioners and give the comparison theorems between the preconditioned and original methods. These results show that the preconditioned GAOR methods converge faster than the GAOR method whenever the GAOR method is convergent. In Section 4, we give an example to confirm our theoretical results.

2. Preliminaries

We need the following definition and results.

Definition 2.1. Let and . We say if for all .
This definition can be carried over to vectors by identifying them with matrices.
In this paper, denotes the spectral radius of a matrix.

Lemma 2.2 (see [6]). Let be nonnegative and irreducible. Then(i)has a positive real eigenvalue equal to its spectral radius ;(ii)for , there corresponds an eigenvector .

Lemma 2.3 (see [7]). Let be nonnegative and irreducible. If for some nonnegative vector , then and is a positive vector.

3. Comparison Results

We consider the preconditioned linear system where and with is a matrix with .

We take as follows:

Now, we obtain two preconditioned linear systems with coefficient matrices where

We split () as then the preconditioned GAOR methods for solving (3.1) are defined as follows where are iteration matrices and

Now, we give comparison results between the preconditioned GAOR methods defined by (3.7) and the corresponding GAOR method defined by (1.6).

Theorem 3.1. Let be the iteration matrices associated with the GAOR and preconditioned GAOR methods, respectively. If the matrix in (1.2) is irreducible with , , , , , , , for some , when , or when , , , then either or

Proof. By direct operation, we have
Since , , , , , , then and is nonnegative. Since is irreducible, from (3.13), it is easy to see that the matrix is nonnegative and irreducible.
Similarly, we can prove that the matrix is a nonnegative and irreducible matrix.
By Lemma 2.2, there is a positive vector such that where . Since the matrix is nonsingular, . Hence, we get either or .
Now, from (3.15) and by the definitions of and , we have
Since , for some , when (), or when , , (), then and . So we have , .
If , then , .
By Lemma 2.3, the inequality (3.11) is proved.
If , then , .
By Lemma 2.3, the inequality (3.12) is proved.

Theorem 3.2. Let be the iteration matrices associated with the GAOR and preconditioned GAOR methods, respectively. If the matrix in (1.2) is irreducible with , , , , , , , for some , when , , or when , , , , then either or
By the analogous proof of Theorem 3.1, we can prove Theorem 3.2.

4. Numerical Example

Now, we present an example to illustrate our theoretical results.

Example 4.1. The coefficient matrix in (1.2) is given by where , , , and with
Table 1 displays the spectral radii of the corresponding iteration matrices with some randomly chosen parameters , , . The randomly chosen parameters and satisfy the conditions of two theorems.
From Table 1, we see that these results accord with Theorems 3.1-3.2.

tab1
Table 1: The spectral radii of the GAOR and preconditioned GAOR iteration matrices.

Acknowledgments

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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