Journal of Applied Mathematics

Volume 2013 (2013), Article ID 827826, 4 pages

http://dx.doi.org/10.1155/2013/827826

Research Article

Comparison Theorems for Single and Double Splittings of Matrices

Cui-Xia Li, Qun-Fa Cui, and Shi-Liang Wu

School of Mathematics and Statistics, Anyang Normal University, Anyang 455000, China

Received 5 February 2013; Accepted 14 March 2013

Academic Editor: Giuseppe Marino

Copyright © 2013 Cui-Xia Li 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.

Some comparison theorems for the spectral radius of double splittings of different matrices under suitable conditions are presented, which are superior to the corresponding results in the recent paper by Miao and Zheng (2009). Some comparison theorems between the spectral radius of single and double splittings of matrices are established and are applied to the Jacobi and Gauss-Seidel double SOR method.

1. Introduction

Consider the linear system

where is nonsingular, is given, and is unknown. The splitting of the coefficient matrix

where is nonsingular, is called a single splitting of in [1]; the basic iterative method for solving (1) is

where matrix is the iteration matrix in (3). Obviously, the iterative method (3) converges to the unique solution of the linear system (1) if and only if the spectral radius of the iteration matrix is smaller than 1. The spectral radius of the iteration matrix is decisive for the convergence and stability, and the smaller it is, the faster the iterative method converges when the spectral radius is smaller than 1. So far, many comparison theorems of single splitting of matrices have been arisen in some papers and books [2–8].

The double splitting of was introduced by Woźnicki [1] and can be described as follows. Splitting the matrix in the form

is called the double splitting of , where is a nonsingular matrix; the corresponding iterative scheme is spanned by three successive iterations:

Following the idea of Golub and Varga [9], Woźnicki wrote (5) in the following equivalent form:

where is the identity matrix. Then, the iterative method (6) converges to the unique solution of (1) for all initial vectors , if and only if the spectral radius of the iteration matrix

is less than one, that is, .

Recently, some comparison theorems for double splittings of monotone matrices and Hermitian positive definite matrices were presented in [8, 10–13]. Elsner et al. [14] presented some comparison theorems of single splittings of different monotone matrices, that is, matrices with nonnegative inverses. Our basic purpose here is to derive some new comparison theorems for the spectral radius of double splittings of different matrices. Under suitable conditions, new comparison theorems are superior to the corresponding results in the recent paper [12]. Some comparison theorems between the spectral radius of single and double splittings of matrices are also established and are applied to the Jacobi and Gauss-Seidel double SOR method.

2. Preliminaries

For convenience, we give some of the notations, definitions, and lemmas which will be used in the sequel.

The matrix is called nonnegative and denoted by if for . We write if for . The matrix is called a monotone matrix if . Matrix is an -matrix if and for all ; .

Definition 1. Let be a nonsingular matrix. Then, the double splitting is

(i)convergent if and only if ; (ii)a regular double splitting if , and ; (iii)a weak regular double splitting if , , and ; (iv)a nonnegative splitting if and .

Lemma 2 (see [3]). Let . Then, , implies and , implies .

Lemma 3 (see [10]). Let and be a weak regular double splitting. Then, .

3. Comparison Theorem

In [12], Miao and Zheng gave a comparison theorem for the spectral radius of double splittings of different monotone matrices. That is, [12, Theorem 3.1] is a major result and is described as follows.

Let and be two monotone matrices, and let , and let be double splittings of and , respectively. Consequently,

Theorem 4 (see [12]). Let and be two nonsingular matrices with and , , and let be weak regular double splittings. If and , then .

Based on the forms of and , we have the following theorem.

Theorem 5. Let and be two nonsingular matrices, and let and be nonnegative splittings. If and , then for .

Proof. Obviously, if and , then . Therefore, we obtain that for .

Based on Definition 1, we obtain the following Theorem 6, which is superior to Theorem 4 [12].

Theorem 6. Let and be two nonsingular matrices, and let and be nonnegative splittings. If and , then for .

Proof. Obviously, and . By the Perron-Frobenius theorem [3], there exists a vector

such that ; that is,

Then, we have

From Lemma 2, we obtain that for .

By investigating Theorem 6, it is easy to see that the conditioners, Theorem 6 are weaker than those of Theorem 4 [12]. That is, the result of Theorem 6 holds without and .

Similarly, we have the following result.

Theorem 7. Let and be two nonsingular matrices, and let and be nonnegative splittings. If and , then for .

4. Convergence for the Jacobi and Gauss-Seidel Double SOR Method

To establish some comparison theorems between the spectral radius of single and double splittings of matrices, based on (3) and (5), we obtain that and . Here and now, .

The result for comparing with is stated as in the following theorem.

Theorem 8. Let be a nonnegative splitting. Then,

(1) for ; (2) for .

Proof. By Definition 1, obviously, matrix . Based on the Perron-Frobenius theorem [3], there exists a vector

such that , that is,

The above equation is equivalent to

From (15), we get that . Substituting it into (14) yields

If , then

That is, . By Lemma 2, it is easy to obtain that .

Obviously, we also obtain that for .

Example 9. Let

Then,

By the simple computations, we have and . Clearly, holds.

Let the matrix be split as

where , and , are strictly lower and upper triangular matrices, respectively, for and . Let

Then, the iterative method (5) corresponding to the double splitting

is called the Jacobi double SOR method [1, 15].

Based on (21), we have the following lemma.

Lemma 10. Let A be an L-matrix, and let the double splittings be defined by (21) and (22). Suppose

Then, the double splitting defined by (22) is regular.

Let

Then, we have the following result.

Theorem 11. Under the conditions of Lemma 10, then

(1) for ; (2) for .

Proof. From Theorem 8, it is easy to see that Theorem 11 holds.

Let

Then, the iterative method (5) corresponding to the double splitting

is called the Gauss-Seidel double SOR method [1, 15].

Let

Similarly, we have the following result.

Theorem 12. Let A be an L-matrix, and let the double splittings be defined by (25) and (26), for and ; then

(1) for ; (2) for .

From Theorems 8, 11, and 12, it is easy to see that the spectral radius of single splitting method is less than the spectral radius of double splitting method under suitable conditions. That is, the efficiency of the single splitting method maybe be superior to that of the double splitting method under suitable conditions.

Acknowledgment

This research was supported by NSFC Tianyuan Mathematics Youth Fund (11026040, 11226337), Science & Technology Development Plan of Henan Province (no. 122300410316) and by Natural Science Foundations of Henan Province (no. 13A110022).

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