Comparison Theorems of Spectral Radius for Splittings of Matrices

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.


Introduction
Let us consider the following linear system: where  ∈ R × is a nonsingular matrix,  ∈ R ×1 is a given vector, and  ∈ R ×1 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  =  −1  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][2][3][4][5][6][7][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  0 ,  1 if and only if the spectral radius of the iteration matrix is less than one; that is, () < 1. 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.

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

Comparison Theorems
Let be two double splittings of .Then, we define Let Then, A ∈ R 2×2 and This shows that A is nonsingular whenever  is nonsingular.
Let A 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.
In [4], they claimed that A is nonsingular whenever  is nonsingular.In fact, we make use of the following strategy to make A nonsingular.That is to say, Obviously, if  − M −1 N is nonsingular, then we immediately obtain that matrix A 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  = M −1 N is less than one.In this case, the iteration scheme ( 6) is convergent.In this meanwhile, we also know that A is nonsingular.Further, comparison theorems discussed are more meaningful as the spectral radius () of the iteration matrix  = M −1 N is less than one.Based on this idea, we can consider the choice of matrixes M and N as In light of this choice, we also have the same as the iteration matrix , In this case, the matrix A is not Then, we have Based on Lemma 4, we have the following results.
Proof.For  = 1, 2, let Then Since That is, from Lemma 4, the results in Theorem 9 hold true.

Theorem 10.
Let be nonnegative and convergent.If either be nonnegative.By direct operation, we obtain From Lemma 5, the results of Theorem 10 hold true.
Obviously, from Lemma 5, we have the following result.

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

Example 1. Assume that
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  ( 1 ) = 0.7949,  ( 2 ) = 0.8195.