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Journal of Applied Mathematics
Volume 2013 (2013), Article ID 827826, 4 pages
Comparison Theorems for Single and Double Splittings of Matrices
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.
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.
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 ; 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  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 , 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.  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 . 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.
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 ). Let . Then, , implies and , implies .
Lemma 3 (see ). Let and be a weak regular double splitting. Then, .
3. Comparison Theorem
In , 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 ). 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 .
Theorem 6. Let and be two nonsingular matrices, and let and be nonnegative splittings. If and , then for .
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
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 , 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
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.
Let Then, we have the following result.
Theorem 11. Under the conditions of Lemma 10, then (1) for ; (2) for .
Let Similarly, we have the following result.
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.
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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