Comparison Theorems for Single and Double Splittings of Matrices

Some comparison theorems for the spectral radius of double splittings of differentmatrices under suitable conditions are presented, which are superior to the corresponding results in the recent paper byMiao 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.


Introduction
Consider the linear system where  ∈ R × is nonsingular,  ∈ R ×1 is given, and  ∈ R ×1 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  =  −1  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 ( −1 ) 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][3][4][5][6][7][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  0 ,  1 if and only if the spectral radius of the iteration matrix is less than one, that is, () < 1.
Recently, some comparison theorems for double splittings of monotone matrices and Hermitian positive definite matrices were presented in [8,[10][11][12][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.

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

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.
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  −1 1 ≥ 0 and  −1 2 ≥ 0. Similarly, we have the following result.Theorem 7. Let  1 and  2 be two nonsingular matrices, and

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,  =  −1 ( + ).
Let the matrix  be split as 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.Then, the double splitting defined by (22) is regular.
Then, we have the following result.
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.
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.