#### Abstract

Based on the methods presented by Song and Yuan (1994), we construct relaxed matrix parallel multisplitting chaotic generalized USAOR-style methods by introducing more relaxed parameters and analyze the convergence of our methods when coefficient matrices are *H*-matrices or irreducible diagonally dominant matrices. The parameters can be adjusted suitably so that the convergence property of methods can be substantially improved. Furthermore, we further study some applied convergence results of methods to be convenient for carrying out numerical experiments. Finally, we give some numerical examples, which show that our convergence results are applied and easily carried out.

#### 1. Introduction

For solving the large sparse linear system where is an real nonsingular matrix and , an iterative method is usually considered. The concept of matrix multisplitting solution of linear system was introduced by O’Leary and White [1] and further studied by many other authors. Frommer and Mayer [2] studied extrapolated relaxed matrix multisplitting methods and matrix multisplitting SOR method. Mas et al. [3] analyzed nonstationary extrapolated relaxed and asynchronous relaxed methods. Gu et al. [4, 5] further studied relaxed nonstationary two-stage matrix multisplitting methods and the corresponding asynchronous schemes.

As we know, matrix multisplitting iterative method for linear systems takes two basic forms. When all of the processors wait until they are updated with the results of the current iteration, it is synchronous. That is to say, when they begin the next iteration of asynchronous, they may act more or less independently of each other and use possibly delayed iterative values of the output of the other processors. In view of the potential time saving inherent in them, asynchronous iterative methods, or chaotic as they are often called, have attracted much attention since the early paper of Chazan and Miranker [6] introduced them in the context of point iterative schemes. Naturally, a number of convergence results [7–21] have been obtained. Recently, the convergence of three relaxed matrix multisplitting chaotic AOR methods has been investigated in [11, 13, 14].

A collection of triples , , is called a multisplitting of if is a splitting of for , and ’s, called weighting matrices, are nonnegative diagonal matrices such that

The unsymmetric accelerated overrelaxation (USAOR) method was proposed in [22], if the diagonal elements of the matrix are nonzero. Without the loss of generality, let the matrix be split as where , , and are nonsingular diagonal, strictly lower triangular and upper triangular parts of . The iterative scheme of the USAOR method is defined by that is, where and are real parameters.

In this paper, we will extend the point USAOR iterative method to matrix multisplitting chaotic generalized USAOR method and analyze their convergence for -matrices and irreducible diagonally dominant matrices.

Based on matrix multisplitting chaotic generalized AOR method [23], the matrix multisplitting chaotic generalized USAOR method is given by where is an iteration matrix and , with , and , real parameters.

*Remark 1. *For , , then GUSAOR method reduces to GAOR method [11]; For , , , , the GUSAOR method reduces to the USAOR method [22].

The remainder of this paper is organized as follows. In Section 2, we introduce some notations and preliminaries. In Section 3, we present relaxed matrix parallel multisplitting chaotic GUSAOR-style methods for solving large nonsingular system, when the coefficient matrix is an -matrix or irreducible diagonally dominant matrix, and analyze the convergence of our methods. In Section 4, we further study some applied convergence results of methods to be convenient for carrying out numerical experiments. In Section 5, we give some numerical examples, which show that our convergence results are easily carried out. Finally, we draw some conclusions.

#### 2. Notation and Preliminaries

We will use the following notation. Let be an matrix. By diag() we denote the diagonal matrix coinciding in its diagonal with . For , , we write if holds for all . Calling * nonnegative* if , we say that if and only if . These definitions carry immediately over to vectors by identifying them with matrices. By we define the absolute value of . We denote by the* comparison matrix* of where for and for , .* Spectral radius* of a matrix is denoted by . It is well known that if and there exists a vector such that , then .

*Definition 2 (see [24]). *Let . It is called an(1)*L*-matrix if for , and for , ;(2)-matrix if it is a nonsingular -matrix satisfying ;(3)-matrix if is an -matrix.

Lemma 3 (see [13]). *If is an -matrix, then *(1)*;
*(2)*there exists a diagonal matrix whose diagonal entries are positive such that with .*

Lemma 4 (see [13]). *Let be an -matrix and let the splitting be an -splitting. If is the diagonal matrix defined in Lemma 3, then .*

Finally, a sequence of sets with is* admissible* if every integer appears infinitely often in the , while such an admissible sequence is regulated if there exists a positive integer such that each of the integers appears at least once in any consecutive sets of the sequence.

#### 3. Relaxed Matrix Multisplitting Parallel Chaotic Methods

Using the given models in [7, 11, 13] and (4), we may describe the corresponding three algorithms of relaxed matrix multisplitting chaotic GUSAOR-style methods, which are as follows.

*Algorithm 5 (given the initial vector). *For , until convergence. Parallel computing the following iterative scheme
with , , , , and , where is the th composition of the affine mapping satisfying

*Remark 6. *In Algorithm 5, by using a suitable positive relaxed parameter , then we can get the following relaxed Algorithm 7.

*Algorithm 7 (given the initial vector). *For , until convergence. Parallel computing the following iterative scheme
with , , , , , and , where is defined such as Algorithm 5.

*Remark 8. *In Algorithm 7, we assume that the index sequence is admissible and regulated; then we can get the following Algorithm 9 with the case of relaxed chaotic GUSAOR method.

*Algorithm 9 (given the initial vector). *For , until convergence. Parallel computing the following iterative scheme
with , , , , , , and , where of Algorithm 5 are replaced by and .

*Remark 10. *Relaxed matrix multisplitting chaotic GUSAOR-style methods introduce more relaxed factors, so our methods are the generalization of [5, 12, 13]. The parameters can be adjusted suitably so that the convergence property of method can be substantially improved.

Using Lemmas 3 and 4, we can get the following convergence result according to Algorithm 5.

Theorem 11. *Let be an -matrix and , , a multisplitting of . Assume that for , we have the following.*(1)* are the strictly lower triangular matrices and are the matrices such that the equalities hold.*(2)*, where .**Then the sequence generated by Algorithm 5 converges for any initial if and only if , , , where
*

*Proof. *Define the iteration matrix in Algorithm 5
where
Obviously, we have to find a constant with and some norm, which are independent of , such that for , .

We first note that the matrices and are -matrices for . Thus by Lemma 3 we have the following inequalities:
From this relation, it follows that
*Case **1*. Let , , . Then
Define
So, for , we have the following relations

Since for , and are both -matrices and , , the splittings and are -splittings of and , respectively, which are -matrices. So, from Lemma 4 we may complete the proving courses, which are listed subsequently.*Case **2*. Let , , .

Assume that , . Then we have
Similar to* Case **1*, we define
Then

It is clear, and are both -matrices. Since, for , and are -matrices, and , the splittings and are -splittings of the matrices and , respectively.

Thus, for* Cases 1* and *2*, from Lemma 4, we have
So
which implies
If we define
Then we have

According to Algorithm 7, we can also get the following result.

Theorem 12. *Let be an -matrix and , , a multisplitting of . Assume that for , we have the following.*(1)* are the strictly lower triangular matrices and are the matrices such that the equalities hold.*(2)*, where .*(3)* is diagonal matrix defined in Lemma 3 and , , , and in Theorem 11.**Then the sequence generated by Algorithm 7 converges for any initial if and only if , , with and , where
*

*Proof. *Define the iteration matrix in Algorithm 7
where
Obviously, similar to Theorem 11 we only need to find a constant with , which is independent of , such that . From the proof of Theorem 11, we can get
Now let denote the last item in above inequalities. Clearly, if , , , then

Using the proving process of Theorems 11 and 12 and [13, Theorem 2.8] we have the following convergence result about Algorithm 9.

Theorem 13. *Let be an -matrix and , , a multisplitting of . Assume that for , we have the following.*(1)* are the strictly lower triangular matrices and are the matrices such that the equalities hold.*(2)*, where .*(3)* is diagonal matrix defined in Lemma 3 and , , and in Theorem 11.*(4)*The index sequence is admissible and regulated.**Then the sequence generated by Algorithm 9 converges for any initial if and only if , , with and , where
*

*Remark 14. *It is known that an -matrix or a symmetric positive definite -matrix is also -matrix. Therefore, the convergence results in Theorems 11, 12, and 13 are valid.

*Remark 15. *Since a strictly or an irreducible diagonally dominant matrix is also satisfying the condition of Theorems 11, 12, and 13, our methods are also valid for these kinds of matrices. Furthermore, for strictly or irreducible diagonally dominant matrices, can take the place of in Theorems 11, 12, and 13.

#### 4. Applied Convergence of Relaxed Chaotic Methods

In Theorems 12 and 13 of this paper, we find that is difficult to compute when carrying out numerical experiments. So for irreducible diagonally dominant matrices, we also have the following applied results of convergence according to Algorithms 7 and 9, respectively.

Theorem 16. *Let be an irreducible diagonally dominant matrix and , , a multisplitting of . Assume that for , we have the following.*(1)* are the strictly lower triangular matrices and are the matrices such that the equalities hold.*(2)*, where .*(3)* is diagonal matrix defined in Lemma 3 and , , , and in Theorem 11.**Then the sequence generated by Algorithm 7 converges for any initial if and only if , , with , where
*

*Proof. *We only prove ). Let us define the iteration matrix in Algorithm 7
From the proving process of Theorem 11, we may define
At first, we need to prove . Similarly, we may also prove . Assume that
With (36), we know that is nonnegative, so according to [24, Theorem 2.7], there exists an eigenvector , , such that hold; that is,
Multiplying by , it holds that
As , it follows by [5, Theorem 11] that
From the proof of [3, Theorem 3.1], we have , . Since the matrices , , and are irreducible, by [24, Theorem 2.1], it follows that
With (40) and (41) and being irreducible diagonally dominant by rows, we have
Notice that if , we may obtain
While if , we also have
From (42), (43), and (44), we can get
From Theorem 11, we have
From (35) and the above proof, we have
where , , .

*Remark 17. *Obviously, convergence results of Theorem 16 are convenient for carrying out numerical experiments. Using the proving process of Theorems 11 and 16 and [13, Theorem 2.8], we can get the following result about Algorithm 9.

Theorem 18. *Let be an irreducible diagonally dominant matrix and , , a multisplitting of . Assume that for , we have the following.*(1)* are the strictly lower triangular matrices and are the matrices such that the equalities hold.*(2)*, where .*(3)* is diagonal matrix defined in Lemma 3 and , , , and in Theorem 11.*(4)*The index sequence is admissible and regulated.**Then the sequence generated by Algorithm 9 converges for any initial if and only if , , with , where
*

*Remark 19. *As a special case, for the relaxed matrix multisplitting chaotic GSSOR-style methods, we have the corresponding convergence results, where , , , with , and real parameters.

#### 5. Numerical Examples

*Example 1. *By using difference discretization of partial differential equation, we can obtain the corresponding coefficient matrix of the linear system , which is as follows:

Now, we will apply the results of Theorem 16 according to Algorithm 7. From Algorithm 7, we can get the iterative matrix Here, we assume that , . By direct calculations with Matlab 7.1, we have In Table 1, we show that convenience results of Section 4 are convenience for carrying out numerical experiments, where denote the spectral radius of iterative matrix :

*Remark 20. *Obviously, of Theorems 16 and 18 is applied and easily calculated when carrying out numerical experiments.

*Example 2. *Consider a matrix of the form
where , is a real number, satisfying .

In Theorem 11, let , , , , ; then Algorithm 5 converges for any initial vector . In Theorems 12 and 13, let , , , , , ; then Algorithms 7 and 9 converge for any initial vector . In Theorems 16 and 18, let , , , , , ; then Algorithms 7 and 9 converge for any initial vector , where .

If we choose , in Theorem 11, let , , , , ; then Algorithm 5 converges for any initial vector . In Theorems 12 and 13, let , , , , , ; then Algorithms 7 and 9 converge for any initial vector . In Theorems 16 and 18, let , , , , , ; then Algorithms 7 and 9 converge for any initial vector , where .

#### 6. Conclusions

In this paper, we consider relaxed matrix parallel multisplitting chaotic GUSAOR-style methods for solving linear systems of algebraic equations , in which the coefficient is an -matrix or an irreducible diagonally dominant matrices, and analyze the convergence of our methods, which use more relaxed factors and are the generalization of [11, 13, 14]. The parameters can be adjusted suitably so that the convergence property of method can be substantially improved. Furthermore, we further study some applied convergence results of methods to be convenient for carrying out numerical experiments. Finally, we give some applied examples, which show that our convergence results are applied and easily calculated when carrying out numerical experiments.

Particularly, one may discuss how to choose the set of relaxed parameters in order to really accelerate the convergence of the considered method. Furthermore, The optimal choice of this set of relaxed parameters is valuably studied.

#### Conflict of Interests

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

#### Acknowledgments

This research of this author is supported by NSFC Tianyuan Mathematics Youth Fund (11226337), NS-FC (61203179, 61202098, 61170309, 91130024, 61033009, 61272544, and 11171039), Aeronautical Science Foundation of China (2013ZD55006), Project of Youth Backbone Teachers of Colleges and Universities in Henan Province (2013GGJS-142), ZZIA Innovation team fund (2014TD02), Major project of development foundation of science and technology of CAEP (2012A0202008), Basic and Advanced Technological Research Project of Henan Province (132300410373, 142300410333), and Natural Science Foundation of Henan Province (14B110023).