Comparison Theorems of Spectral Radius for Splittings of Matrices

Abstract

A class of the iteration method from the double splitting of coefficient matrix for solving the linear system is further investigated. By structuring a new matrix, the iteration matrix of the corresponding double splitting iteration method is presented. On the basis of convergence and comparison theorems for single splittings, we present some new convergence and comparison theorems on spectral radius for splittings of matrices.

1. Introduction

Let us consider the following linear system: where is a nonsingular matrix, is a given vector, and is an unknown vector. In order to solve the linear system (1) by iteration methods, the coefficient matrix is split into where is nonsingular; then, an iterative formula for solving the linear system (1) is where is the iteration matrix in (3).

The splitting (2) is called a (single) splitting of and the iteration method (3) is called a (one-step) linear stationary iteration method. Obviously, the iteration 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 iteration method converges when the spectral radius is smaller than 1. So far, many comparison theorems of single splittings of matrices have been presented in some papers and books [1–8].

Woźnicki [9] introduced the double splitting of as where is a nonsingular matrix. The corresponding iterative scheme is spanned by three successive iterations: which can be rewritten in the equivalent form where is the identity matrix. The iteration method given by (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 convergence and comparison results for double splittings of matrices are presented. In [10], some convergence theorems for the double splitting of a monotone matrix or a Hermitian positive definite matrix are presented. Compared with the results in [10], some improved convergence and comparison results for the double splitting of a Hermitian positive definite matrix are proposed in [11]. In [12], some convergence results for the double splitting of a non-Hermitian positive semidefinite matrix are established. Further, some comparison theorems for double splittings of different monotone matrices are given in [13, 14] and some convergence and comparison results for nonnegative double splittings of matrices are given in [4, 15]. In this paper, by structuring a new matrix, the iteration matrix of the corresponding iteration method from double splitting of coefficient matrix is presented. On the basis of convergence and comparison theorems for single splittings, we present some new convergence and comparison theorems on spectral radius for splittings of matrices.

2. Preliminaries

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

The matrix is called nonnegative and is denoted by if for , . We write    if for , . The matrix is called a monotone matrix if .

Definition 1. Let be a nonsingular matrix. Then, is called (i)regular if and ;(ii)weak regular if and ;(iii)nonnegative if ;(iv)-splitting if is an -matrix and .

Definition 2 (see [4, 10, 15]). 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 ;(v)an -double splitting if is an -matrix and and .

Lemma 3 (see [2]). Let . Then

Lemma 4 (see [16]). Let and be -splittings of (i.e., are -matrices; , ) and Then, exactly one of the following statements holds: (1). In addition, if is irreducible, the first inequality is also strict;(2);(3).

Lemma 5 (see [17]). Let and be nonnegative and convergent. (1)If either or , then .(2)If there exist , , such that or , then .

3. Comparison Theorems

Let be two double splittings of . Then, we define Let Then, and This shows that is nonsingular whenever is nonsingular. Let be split as with Then

In [4], some comparison theorems for the double splitting (4) through investigating the matrix splitting defined by (14) were obtained, which were described as follows.

Theorem 6 (see [4]). Let , and let the two double splittings (10) be nonnegative and convergent. Suppose then

Corollary 7 (see [4]). Let , and let the two double splittings (10) be nonnegative and convergent. Suppose then

Theorem 8 (see [4]). Let , be regular double splitting, and let be nonnegative and convergent double splitting. Suppose then .

In [4], they claimed that is nonsingular whenever is nonsingular. In fact, we make use of the following strategy to make nonsingular. That is to say, Obviously, if is nonsingular, then we immediately obtain that matrix is nonsingular too. When one discusses the convergence properties of the iteration scheme (6), it is expected that the spectral radius of the iteration matrix is less than one. In this case, the iteration scheme (6) is convergent. In this meanwhile, we also know that is nonsingular. Further, comparison theorems discussed are more meaningful as the spectral radius of the iteration matrix is less than one. Based on this idea, we can consider the choice of matrixes and as In light of this choice, we also have the same as the iteration matrix , In this case, the matrix is not but is Then, we have

Based on Lemma 4, we have the following results.

Theorem 9. Let be M-splittings of . If , then exactly one of the following statements holds: (1). In addition, if is irreducible, the first inequality is also strict;(2);(3).

Proof. For , let Then Since , then . That is, from Lemma 4, the results in Theorem 9 hold true.

Theorem 10. Let be nonnegative and convergent. If either or , then .

Proof. Let be nonnegative. By direct operation, we obtain Since or , we have or . From Lemma 5, the results of Theorem 10 hold true.

Obviously, from Lemma 5, we have the following result.

Theorem 11. Let be nonnegative and convergent. If there exist , , such that or , then .

Compared with Corollary 7 and Theorem 8, the condition in Theorems 9, 10, and 11 is not necessary.

Theorem 12. Let be nonnegative and convergent. If , then .

Proof. Obviously, if , then . From Theorem  2.11 in [17], the results of Theorem 12 hold true.

Theorem 13. Let let be regular, and let be nonnegative and convergent. If , then .

Proof. Assume that ; the result is trivial. Assume that . Then, the splitting is nonnegative.
By the Perron-Frobenius Theorem in [1], there exists a vector and , such that . From , we derive . Since , then . From Theorem 2.17 in [17], the results of Theorem 13 hold true.

4. Numerical Example

In this section, we make use of an example to illustrate Theorems 9, 10, 12, and 13.

Example 1. Assume that Then Let Then Therefore, we have the following facts. That is to say, which, respectively, satisfy the conditions of Theorems 9, 10, 12, and 13. In this case, by the simple computations, we have Clearly, . That is to say, Theorems 9, 10, 12, and 13 hold true.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This research was supported by NSFC (no. 11301009), by Science & Technology Development Plan of Henan Province (no. 122300410316), and by Natural Science Foundations of Henan Province (no. 13A110022).