Convergence of Relaxed Matrix Parallel Multisplitting Chaotic Methods for 𝐻-Matrices

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 𝐴𝑥=𝑏, (1) where 𝐴 is an 𝑛×𝑛 real nonsingular matrix and 𝑥,𝑏∈𝑅^𝑛, 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 (𝑀_𝑙,𝑁_𝑙,𝐸_𝑙), 𝑙=1,2,…,𝑠, is called a multisplitting of 𝐴 if 𝐴=𝑀_𝑙−𝑁_𝑙 is a splitting of 𝐴 for 𝑙=1,2,…,𝑠, and 𝐸_𝑙’s, called weighting matrices, are nonnegative diagonal matrices such that ∑^𝑠_𝑙=1𝐸_𝑙=𝐼.

The unsymmetric accelerated overrelaxation (USAOR) method was proposed in [22], if the diagonal elements of the matrix 𝐴 are nonzero. Without the loss of generality, let the matrix 𝐴 be split as 𝐴=𝐼−𝐿−𝑈, (2) where 𝐼, 𝐿, and 𝑈 are nonsingular diagonal, strictly lower triangular and upper triangular parts of 𝐴. The iterative scheme of the USAOR method is defined by 𝑥^(𝑘+(1/2))=𝐻𝜔₁,𝜔₂𝑥^(𝑘)+𝜔₁𝐼−𝜔₂𝐿⁻¹𝑥𝑏,_(𝑘+1)=𝐻𝜔₃,𝜔₄𝑥^(𝑘+(1/2))+𝜔₃𝐼−𝜔₄𝑈𝑏; (3) that is, 𝑥^(𝑘+1)=𝐻_USAOR𝑥^(𝑘)+𝐺_USAOR,𝑘=0,1,…, (4) where 𝐻_USAOR=𝐻𝜔₃,𝜔₄𝐻𝜔₁,𝜔₂,𝐻𝜔₁,𝜔₂=𝐼−𝜔₂𝐿⁻¹×1−𝜔₁𝐼+𝜔₁−𝜔₂𝐿+𝜔₁𝑈,𝐻𝜔₃,𝜔₄=𝐼−𝜔₄𝑈⁻¹1−𝜔₃𝐼+𝜔₃−𝜔₄𝑈+𝜔₃𝐿,𝐺_USAOR=𝐼−𝜔₄𝑈⁻¹𝜔₁+𝜔₃−𝜔₁𝜔₃𝐼+𝜔₃𝜔₁−𝜔₂𝐿+𝜔₁𝜔₃−𝜔₄𝑈𝐼−𝜔₂𝐿⁻¹𝑏, (5) and 𝛼_𝑖,𝛽_𝑖(𝑖=1,2) 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 𝐻-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 𝑥^(𝑘+1)=𝐻_GUSAOR𝑥^(𝑘)+𝐺_GUSAOR,𝑘=0,1,2,…, (6) where 𝐻_GUSAOR𝐻,𝐻==𝐻𝛽,𝜓𝛼,𝜙𝛼,𝜙𝐼−𝛼𝜙𝐿⁻¹()𝐻=𝐼−𝜙+1−𝛼𝜙𝐿+𝜙𝑈𝛽,𝜓𝐼−𝛽𝜓𝑈⁻¹,𝐺𝐼−𝜓+1−𝛽𝜓𝑈+𝜓𝐿_GUSAOR=1−𝛽𝜓𝑈⁻¹𝐿+2𝐼−𝜓+𝜓−𝛼𝜙1−𝛽𝜓𝑈𝐼−𝛼𝜙𝐿⁻¹𝑏, (7) is an iteration matrix and 𝜙=diag(𝛽₁,𝛽₂,…,𝛽_𝑛), 𝜓=diag(𝛾₁,𝛾₂,…,𝛾_𝑛) with 𝛽_𝑖≥0, 𝛾_𝑖≥0(𝑖=1,2,…,𝑛) and 𝛼, 𝛽 real parameters.

Remark 1. For 𝛽=𝛾_𝑖=0, 𝑖=1,2,…,𝑛, then GUSAOR method reduces to GAOR method [11]; For 𝜙=𝜔₁𝐼, 𝛼=𝜔₂/𝜔₁, 𝜓=𝜔₃𝐼, 𝛽=𝜔₄/𝜔₃, 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 𝐴 is an 𝐻-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 𝐶=(𝑐_𝑖𝑗)∈𝑅^𝑛×𝑛 be an 𝑛×𝑛 matrix. By diag(𝐶) we denote the diagonal matrix coinciding in its diagonal with 𝐶. For 𝐴=(𝑎_𝑖𝑗), 𝐵=(𝑏_𝑖𝑗)∈𝑅^𝑛×𝑛, we write 𝐴≥𝐵 if 𝑎_𝑖𝑗≥𝑏_𝑖𝑗 holds for all 𝑖,𝑗=1,2,…,𝑛. Calling 𝐴 nonnegative if 𝐴≥0, we say that 𝐵≤𝐶 if and only if −𝐵≥−𝐶. These definitions carry immediately over to vectors by identifying them with 𝑛×1 matrices. By |𝐴|=(|𝑎_𝑖𝑗|) we define the absolute value of 𝐴∈𝑅^𝑛×𝑛. We denote by ⟨𝐴⟩=(⟨𝑎_𝑖𝑗⟩) the comparison matrix of 𝐴∈𝑅^𝑛×𝑛 where ⟨𝑎_𝑖𝑗⟩=|𝑎_𝑖𝑗| for 𝑖=𝑗 and ⟨𝑎_𝑖𝑗⟩=−|𝑎_𝑖𝑗| for 𝑖≠𝑗, 𝑖,𝑗=1,2,…,𝑛. Spectral radius of a matrix 𝐴 is denoted by 𝜌(𝐴). It is well known that if 𝐴≥0 and there exists a vector 𝑥>0 such that 𝐴𝑥<𝛼𝑥, then 𝜌(𝐴)<𝛼.

Definition 2 (see [24]). Let 𝐴∈𝑅^𝑛×𝑛. It is called an(1)L-matrix if 𝑎_𝑖𝑖>0 for 𝑖=1,2,…,𝑛, and 𝑎_𝑖𝑗≤0 for 𝑖≠𝑗, 𝑖,𝑗=1,2,…,𝑛;(2)𝑀-matrix if it is a nonsingular 𝐿-matrix satisfying 𝐴⁻¹≥0;(3)𝐻-matrix if ⟨𝐴⟩ is an 𝑀-matrix.

Lemma 3 (see [13]). If 𝐴 is an 𝐻-matrix, then (1)|𝐴⁻¹|≤⟨𝐴⟩⁻¹; (2)there exists a diagonal matrix 𝑃 whose diagonal entries are positive such that ⟨𝐴⟩𝑃𝑒>0 with 𝑒=(1,1,…,1)^𝑇.

Lemma 4 (see [13]). Let 𝐴 be an 𝑀-matrix and let the splitting 𝐴=𝑀−𝑁 be an 𝑀-splitting. If 𝑃 is the diagonal matrix defined in Lemma 3, then ‖𝑃⁻¹𝑀⁻¹𝑁𝑃‖_∞<1.

Finally, a sequence of sets 𝑃_𝑖 with 𝑃_𝑖⊆{1,…,𝑠} is admissible if every integer 1,2,…,𝑠 appears infinitely often in the 𝑃_𝑖, while such an admissible sequence is regulated if there exists a positive integer 𝑇 such that each of the integers appears at least once in any 𝑇 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 𝑘=0,1,…, until convergence. Parallel computing the following iterative scheme 𝑥^(𝑘+1)=^𝑠_𝑙=1𝐸_𝑙𝐻_GUSAOR^{𝜇𝑙,𝑘}_𝑙𝑥^(𝑘),𝐻_GUSAOR𝑙𝑥^(𝑘)=𝐻𝛽,𝜓_𝑙𝐻𝛼,𝜙_𝑙𝑥^(𝑘)+𝐺_GUSAOR𝑙𝑏,𝐻(𝛼,𝜙)_𝑙=𝐼−𝛼𝜙𝐿_𝑙⁻¹𝐼−𝜙+(1−𝛼)𝜙𝐿_𝑙+𝜙𝑈_𝑙,𝐻𝛽,𝜓_𝑙=𝐼−𝛽𝜓𝑈_𝑙⁻¹𝐼−𝜓+1−𝛽𝜓𝑈_𝑙+𝜓𝐿_𝑙,𝐺_GUSAOR𝑙=𝐼−𝛽𝜓𝑈_𝑙⁻¹×2𝐼−𝜓+𝜓−𝛼𝜙𝐿_𝑙+1−𝛽𝜓𝑈_𝑙×𝐼−𝛼𝜙𝐿_𝑙⁻¹, (8) with 𝛼≥0, 𝛽≥0, 𝛽_𝑖>0, 𝛾_𝑖>0, and 𝜇_𝑙,𝑘≥1, where (𝐻_GUSAOR)^{𝜇𝑙,𝑘}_𝑙 is the 𝜇_𝑙,𝑘th composition of the affine mapping satisfying 𝐻_GUSAOR^{𝜇𝑙,𝑘}_𝑙=𝐻_GUSAOR𝑙⋯𝐻_GUSAOR𝑙,𝜇_𝑙,𝑘≥1,𝐼,𝜇_𝑙,𝑘=0,𝐵=𝐿_𝑙+𝑈_𝑙,𝑙=1,2,…,𝑠. (9)

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

Algorithm 7 (given the initial vector). For 𝑘=0,1,…, until convergence. Parallel computing the following iterative scheme 𝑥^(𝑘+1)=𝜔^𝑠_𝑙=1𝐸_𝑙𝐻_GUSAOR^{𝜇𝑙,𝑘}_𝑙𝑥^(𝑘)+(1−𝜔)𝑥^(𝑘), (10) with 𝜔>0, 𝛼≥0, 𝛽≥0, 𝛽_𝑖>0, 𝛾_𝑖>0, and 𝜇_𝑙,𝑘≥1, where (𝐻_GUSAOR)^{𝜇𝑙,𝑘}_𝑙(𝑥^(𝑘)) is defined such as Algorithm 5.

Remark 8. In Algorithm 7, we assume that the index sequence {𝑃_𝑖} 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 𝑘=0,1,…, until convergence. Parallel computing the following iterative scheme 𝑥^(𝑘+1)=(𝐼−𝜔)_{𝑙∈𝑃𝑖}𝐸_𝑙+𝜔_{𝑙∈𝑃𝑖}𝐸_𝑙𝐻_GUSAOR^{𝜇𝑙,𝑘}_𝑙𝑥^{(𝑘−𝑟𝑘+1)}, (11) with 𝜔>0, 𝛼≥0, 𝛽≥0, 𝛽_𝑖>0, 𝛾_𝑖>0, 𝜇_𝑙,𝑘≥1, and 𝑃_𝑖⊆{1,…,𝑠}, where (𝐻_GUSAOR)_𝑙(𝑥^(𝑘)) of Algorithm 5 are replaced by (𝐻_GUSAOR)_𝑙(𝑥^{(𝑘−𝑟𝑘+1)}) and 𝑥^{(𝑘−𝑟𝑘+1)}=(𝑥₁^{(𝑘−𝑟(1,𝑘))},𝑥₂^{(𝑘−𝑟(2,𝑘))},…,𝑥_𝑛^{(𝑘−𝑟(𝑛,𝑘))})^𝑇.

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 𝐴∈𝑅^𝑛×𝑛 be an 𝐻-matrix and (𝐼−𝐿_𝑙,𝑈_𝑙,𝐸_𝑙), 𝑙=1,2,…,𝑠, a multisplitting of 𝐴. Assume that for 𝑙=1,2,…,𝑠, we have the following.(1)𝐿_𝑙 are the strictly lower triangular matrices and 𝑈_𝑙 are the matrices such that the equalities 𝐴=𝐼−𝐿_𝑙−𝑈_𝑙 hold.(2)⟨𝐴⟩=|𝐼|−|𝐿_𝑙|−|𝑈_𝑙|=|𝐼|−|𝐵|, where |𝐵|=|𝐿_𝑙|+|𝑈_𝑙|.Then the sequence {𝑥^(𝑘)} generated by Algorithm 5 converges for any initial 𝑥⁽⁰⁾ if and only if (𝛼,𝛽,𝛽_𝑖,𝛾_𝑖), 𝑖=1,2,…𝑛∈𝑊₁, 𝜌=(|𝐵|), where 𝑊₁=𝛼,𝛽,𝛽_𝑖,𝛾_𝑖∈𝑅⁴∶0≤𝛼,𝛽≤1,0<𝛽_𝑖,𝛾_𝑖<2.1+𝜌,𝑖=1,2,…,𝑛 (12)

Proof. Define the iteration matrix in Algorithm 5𝐻_GUSAOR𝑘=^𝑠_𝑙=1𝐸_𝑙𝐻𝛽,𝜓_𝑙𝐻𝛼,𝜙_𝑙^{𝜇𝑙,𝑘}, (13) where 𝐻𝛼,𝜙_𝑙=𝐼−𝛼𝜙𝐿_𝑙⁻¹𝐼−𝜙+(1−𝛼)𝜙𝐿_𝑙+𝜙𝑈_𝑙,𝐻𝛽,𝜓_𝑙=𝐼−𝛽𝜓𝑈_𝑙⁻¹𝐼−𝜓+1−𝛽𝜓𝑈_𝑙+𝜓𝐿_𝑙. (14) Obviously, we have to find a constant 𝜎 with 0≤𝜎<1 and some norm, which are independent of 𝑘, such that for 𝑘≥1, ‖𝐻_GUSAOR(𝛼,𝛽,𝜙,𝜓)_𝑘‖≤𝜎.
We first note that the matrices 𝐼−𝛼𝜙𝐿_𝑙 and 𝐼−𝛽𝜓𝑈_𝑙 are 𝐻-matrices for 𝑙=1,2,…,𝑠. Thus by Lemma 3 we have the following inequalities: 𝐼−𝛼𝜙𝐿_𝑙⁻¹≤𝐼−𝛼𝜙𝐿_𝑙⁻¹=𝐼−𝛼𝜙𝐿_𝑙⁻¹,𝐼−𝛽𝜓𝑈_𝑙⁻¹≤𝐼−𝛽𝜓𝑈_𝑙⁻¹=𝐼−𝛽𝜓𝑈_𝑙⁻¹. (15) From this relation, it follows that 𝐻𝛽,𝜓_𝑙𝐻𝛼,𝜙_𝑙=𝐼−𝛽𝜓𝑈_𝑙⁻¹×𝐼−𝜓+1−𝛽𝜓𝑈_𝑙+𝜓𝐿_𝑙×𝐼−𝛼𝜙𝐿_𝑙⁻¹×𝐼−𝜙+(1−𝛼)𝜙𝐿_𝑙+𝜙𝑈_𝑙≤𝐼−𝛽𝜓𝑈_𝑙⁻¹×𝐼−𝜓+1−𝛽𝜓𝑈_𝑙+𝜓𝐿_𝑙×𝐼−𝛼𝜙𝐿_𝑙⁻¹×𝐼−𝜙+(1−𝛼)𝜙𝐿_𝑙+𝜙𝑈_𝑙≤𝐼−𝛽𝜓𝑈_𝑙⁻¹×𝐼−𝜓+1−𝛽𝜓𝑈_𝑙+𝜓𝐿_𝑙×𝐼−𝛼𝜙𝐿_𝑙⁻¹×𝐼−𝜙+|1−𝛼|𝜙𝐿_𝑙+𝜙𝑈_𝑙. (16)
Case 1. Let 0<𝛽_𝑖≤1, 0<𝛾_𝑖≤1, 𝑖=1,2,…,𝑛. Then 𝐼−𝜓+1−𝛽𝜓𝑈_𝑙+𝜓𝐿_𝑙=𝐼−𝜓+𝜓𝑈_𝑙−𝛽𝜓𝑈_𝑙+𝜓𝐿_𝑙.𝐼−𝜙+|1−𝛼|𝜙𝐿_𝑙+𝜙𝑈_𝑙=𝐼−𝜙+𝜙𝐿_𝑙−𝛼𝜙𝐿_𝑙+𝜙𝑈_𝑙. (17) Define 𝑀¹_𝑙𝛽,𝜓=𝐼−𝛽𝜓𝑈_𝑙,𝑀²_𝑙𝛼,𝜙=𝐼−𝛼𝜙𝐿_𝑙,𝑁¹_𝑙𝛽,𝜓=𝐼−𝜓+1−𝛽𝜓𝑈_𝑙+𝜓𝐿_𝑙,𝑁²_𝑙𝛼,𝜙=𝐼−𝜙+(1−𝛼)𝜙𝐿_𝑙+𝜙𝑈_𝑙. (18) So, for 𝑙=1,2,…,𝑠, we have the following relations 𝑀¹_𝑙𝛽,𝜓−𝑁¹_𝑙𝛽,𝜓=𝜓−𝜓𝑈_𝑙−𝜓𝐿_𝑙(|𝐵.𝑀=𝜓𝐼−|)_2𝑙𝛼,𝜙−𝑁²_𝑙𝛼,𝜙=𝜙−𝜙𝐿_𝑙−𝜙𝑈_𝑙=𝜙(𝐼−|𝐵|). (19)
Since for 𝑙=1,2,…,𝑠, 𝑀¹_𝑙(𝛽,𝜓) and 𝑀²_𝑙(𝛼,𝜙) are both 𝑀-matrices and 𝑁¹_𝑙(𝛽,𝜓)≥0, 𝑁²_𝑙(𝛼,𝜙)≥0, the splittings 𝑀¹_𝑙(𝛽,𝜓)−𝑁¹_𝑙(𝛽,𝜓) and 𝑀²_𝑙(𝛼,𝜙)−𝑁²_𝑙(𝛼,𝜙) are 𝑀-splittings of 𝜓(𝐼−|𝐵|) and 𝜙(𝐼−|𝐵|), respectively, which are 𝑀-matrices. So, from Lemma 4 we may complete the proving courses, which are listed subsequently.
Case 2. Let 1<𝛽_𝑖<2/(1+𝜌), 1<𝛾_𝑖<2/(1+𝜌), 𝑖=1,2,…,𝑛.
Assume that 𝛽_max=max_{1≤𝑖≤𝑛}{𝛽_𝑖}, 𝛾_max=max_{1≤𝑖≤𝑛}{𝛾_𝑖}. Then we have 𝐼−𝜓+1−𝛽𝜓𝑈_𝑙+𝜓𝐿_𝑙≤𝛾_max−1𝐼+1−𝛽𝛾_max𝑈_𝑙+𝛾_max𝐿_𝑙,𝐼−𝜙+(1−𝛼)𝜙𝐿_𝑙+𝜙𝑈_𝑙≤𝛽_max−1𝐼+(1−𝛼)𝛽_max𝐿_𝑙+𝛽_max𝑈_𝑙,𝐼−𝛽𝜓𝑈_𝑙⁻¹≤𝐼−𝛽𝛾_max𝑈_𝑙⁻¹,𝐼−𝛼𝜙𝐿_𝑙⁻¹≤𝐼−𝛼𝛽_max𝐿_𝑙⁻¹. (20) Similar to Case 1, we define 𝑀³_𝑙𝛽,𝛾_max=𝐼−𝛽𝛾_max𝑈_𝑙,𝑀⁴_𝑙𝛼,𝛽_max=𝐼−𝛼𝛽_max𝐿_𝑙,𝑁³_𝑙𝛽,𝛾_max=𝛾_max−1𝐼+1−𝛽𝛾_max𝑈_𝑙+𝛾_max𝐿_𝑙,𝑁⁴_𝑙𝛼,𝛽_max=𝛽_max−1𝐼+(1−𝛼)𝛽_max𝐿_𝑙+𝛽_max𝑈_𝑙. (21) Then 𝑀³_𝑙𝛽,𝛾_max−𝑁³_𝑙𝛽,𝛾_max=|𝐵|,𝑀2−𝛽𝐼−𝛽_4𝑙𝛼,𝛽_max−𝑁⁴_𝑙𝛼,𝛽_max=(2−𝛼)𝐼−𝛼|𝐵|. (22)
It is clear, (2−𝛽)𝐼−𝛽|𝐵| and (2−𝛼)𝐼−𝛼|𝐵| are both 𝑀-matrices. Since, for 𝑙=1,2,…,𝑠, 𝑀¹_𝑙(𝛽,𝜓) and 𝑀²_𝑙(𝛼,𝜙) are 𝑀-matrices, 𝑁¹_𝑙(𝛽,𝜓)≥0 and 𝑁²_𝑙(𝛼,𝜙)≥0, the splittings 𝑀¹_𝑙(𝛽,𝜓)−𝑁¹_𝑙(𝛽,𝜓) and 𝑀²_𝑙(𝛼,𝜙)−𝑁²_𝑙(𝛼,𝜙) are 𝑀-splittings of the matrices (2−𝛽)𝐼−𝛽|𝐵| and (2−𝛼)𝐼−𝛼|𝐵|, respectively.
Thus, for Cases  1 and 2, from Lemma 4, we have 𝑃₁⁻¹𝑀¹_𝑙⁻¹𝛽,𝜓𝑁¹_𝑙𝛽,𝜓𝑃_1∞𝑃<1,𝑙=1,2,…,𝑠,₂₋₁𝑀²_𝑙⁻¹𝛼,𝜙𝑁²_𝑙𝛼,𝜙𝑃_2∞<1,𝑙=1,2,…,𝑠, (23) So 𝑃₁⁻¹𝐻_GUSAOR𝑘𝑃₁𝑒=^𝑠_𝑙=1𝐸_𝑙𝑃₁⁻¹𝐻𝛽,𝜓_𝑙𝐻𝛼,𝜙_𝑙𝑃₁^{𝜇𝑙,𝑘}𝑒≤^𝑠_𝑙=1𝐸_𝑙𝑃₁⁻¹𝑀¹_𝑙⁻¹𝛽,𝜓𝑁¹_𝑙𝛽,𝜓𝑀²_𝑙⁻¹×𝛼,𝜙𝑁²_𝑙𝛼,𝜙𝑃₁^{𝜇𝑙,𝑘}𝑒=^𝑠_𝑙=1𝐸_𝑙𝑃₁⁻¹𝑀¹_𝑙⁻¹𝛽,𝜓𝑁¹_𝑙𝛽,𝜓×𝑃₁𝑃₁⁻¹𝑃₂𝑃₂⁻¹𝑀²_𝑙⁻¹×𝛼,𝜙𝑁²_𝑙𝛼,𝜙𝑃₂𝑃₂⁻¹𝑃₁^{𝜇𝑙,𝑘}𝑒=max_{1≤𝑙≤𝑠}𝑃₁⁻¹𝑀¹_𝑙⁻¹𝛽,𝜓𝑁¹_𝑙𝛽,𝜓𝑃_1∞×𝑃₂⁻¹𝑀²_𝑙⁻¹𝛼,𝜙𝑁²_𝑙𝛼,𝜙𝑃_2∞𝑒. (24) which implies 𝑃₁⁻¹(𝐻_GUSAOR)_𝑘𝑃_1∞≤max_{1≤𝑙≤𝑠}𝑃₁⁻¹𝑀¹_𝑙⁻¹𝛽,𝜓𝑁¹_𝑙𝛽,𝜓𝑃_1∞×𝑃₂⁻¹𝑀²_𝑙⁻¹𝛼,𝜙𝑁²_𝑙𝛼,𝜙𝑃_2∞𝐻<1,_GUSAOR𝑘𝑃₁𝑒=𝑃₁𝑃₁⁻¹𝐻_GUSAOR𝛼,𝛽,𝜙,𝜓_𝑘𝑃₁𝑒≤𝑃₁𝑃₁⁻¹𝐻_GUSAOR(𝛼,𝛽,𝜙,𝜓)_𝑘𝑃_1∞𝑒≤max_{1≤𝑙≤𝑠}𝑃₁⁻¹𝑀¹_𝑙⁻¹𝛽,𝜓𝑁¹_𝑙𝛽,𝜓𝑃_1∞×𝑃₂⁻¹𝑀²_𝑙⁻¹𝛼,𝜙𝑁²_𝑙𝛼,𝜙𝑃_2∞𝑃₁𝑒. (25) If we define 𝜎=max_{1≤𝑙≤𝑠}𝑃₁⁻¹𝑀¹_𝑙⁻¹𝛽,𝜓𝑁¹_𝑙𝛽,𝜓𝑃_1∞×𝑃₂⁻¹𝑀²_𝑙⁻¹𝛼,𝜙𝑁²_𝑙𝛼,𝜙𝑃_2∞. (26) Then we have 𝐻_GUSAOR𝑘≤𝜎<1. (27)

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

Theorem 12. Let 𝐴∈𝑅^𝑛×𝑛 be an 𝐻-matrix and (𝐼−𝐿_𝑙,𝑈_𝑙,𝐸_𝑙), 𝑙=1,2,…,𝑠, a multisplitting of 𝐴. Assume that for 𝑙=1,2,…,𝑠, we have the following.(1)𝐿_𝑙 are the strictly lower triangular matrices and 𝑈_𝑙 are the matrices such that the equalities 𝐴=𝐼−𝐿_𝑙−𝑈_𝑙 hold.(2)⟨𝐴⟩=|𝐼|−|𝐿_𝑙|−|𝑈_𝑙|=|𝐼|−|𝐵|, where |𝐵|=|𝐿_𝑙|+|𝑈_𝑙|.(3)𝑃 is diagonal matrix defined in Lemma 3 and 𝑀¹_𝑙(𝛽,𝜓), 𝑁¹_𝑙(𝛽,𝜓), 𝑀²_𝑙(𝛼,𝜙), and 𝑁²_𝑙(𝛼,𝜙) in Theorem 11.Then the sequence {𝑥^(𝑘)} generated by Algorithm 7 converges for any initial 𝑥⁽⁰⁾ if and only if (𝛼,𝛽,𝛽_𝑖,𝛾_𝑖,𝜔), 𝑖=1,2,…,𝑛∈𝑊₂, with 𝜌=(|𝐵|) and 𝜃=max_{1≤𝑙≤𝑠}{‖𝑃₁⁻¹(𝑀¹_𝑙)⁻¹(𝛽,𝜓)𝑁¹_𝑙(𝛽,𝜓)𝑃₁‖_∞‖𝑃₂⁻¹(𝑀²_𝑙)⁻¹ (𝛼,𝜙)𝑁²_𝑙(𝛼,𝜙)𝑃₂‖_∞}, where 𝑊₂=𝛼,𝛽,𝛽_𝑖,𝛾_𝑖,𝜔∈𝑅⁵∶0≤𝛼,𝛽≤1,0<𝛽_𝑖,𝛾_𝑖<221+𝜌,0<𝜔<.1+𝜃,𝑖=1,2,…,𝑛. (28)

Proof. Define the iteration matrix in Algorithm 7𝐻_GUSAOR𝑘=𝜔𝐻_GUSAOR𝑘+(1−𝜔)𝐼, (29) where 𝐻_GUSAOR𝑘=^𝑠_𝑙=1𝐸_𝑙{𝐻𝛽,𝜓_𝑙𝐻𝛼,𝜙_𝑙}^{𝜇𝑙,𝑘}.𝐻𝛼,𝜙_𝑙=𝐼−𝛼𝜙𝐿_𝑙⁻¹𝐼−𝜙+(1−𝛼)𝜙𝐿_𝑙+𝜙𝑈_𝑙,𝐻𝛽,𝜓_𝑙=𝐼−𝛽𝜓𝑈_𝑙⁻¹𝐼−𝜓+1−𝛽𝜓𝑈_𝑙+𝜓𝐿_𝑙. (30) Obviously, similar to Theorem 11 we only need to find a constant 𝜎 with 0≤𝜎<1, which is independent of 𝑘, such that 𝐾=‖𝑃₁⁻¹𝐻_{𝐺𝑈𝑆𝐴𝑂𝑅}(𝛼,𝛽,𝜙,𝜓,𝜔)_𝑘𝑃₁‖_∞≤𝜎. From the proof of Theorem 11, we can get 𝐾≤𝜔𝑃₁⁻¹𝐻_GUSAOR𝑘𝑃_1∞+|1−𝜔|≤𝜔max_{1≤𝑙≤𝑠}𝑃₁⁻¹𝑀¹_𝑙⁻¹𝛽,𝜓𝑁¹_𝑙𝛽,𝜓𝑃_1∞×𝑃₂⁻¹𝑀²_𝑙⁻¹𝛼,𝜙𝑁²_𝑙𝛼,𝜙𝑃_2∞+||||.1−𝜔=𝜔𝜃+1−𝜔 (31) Now let 𝜎 denote the last item in above inequalities. Clearly, if 0<𝛽_𝑖<2/(1+𝜌), 0<𝛾_𝑖<2/(1+𝜌), 0<𝜔<2/(1+𝜃), then 𝐻_GUSAOR𝑘≤𝜎<1. (32)

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 𝐴∈𝑅^𝑛×𝑛 be an 𝐻-matrix and (𝐼−𝐿_𝑙,𝑈_𝑙,𝐸_𝑙), 𝑙=1,2,…,𝑠, a multisplitting of 𝐴. Assume that for 𝑙=1,2,…,𝑠, we have the following.(1)𝐿_𝑙 are the strictly lower triangular matrices and 𝑈_𝑙 are the matrices such that the equalities 𝐴=𝐼−𝐿_𝑙−𝑈_𝑙 hold.(2)⟨𝐴⟩=|𝐼|−|𝐿_𝑙|−|𝑈_𝑙|=|𝐼|−|𝐵|, where |𝐵|=|𝐿_𝑙|+|𝑈_𝑙|.(3)𝑃 is diagonal matrix defined in Lemma 3 and 𝑀¹_𝑙(𝛽,𝜓), 𝑁¹_𝑙(𝛽,𝜓), 𝑀²_𝑙(𝛼,𝜙) and 𝑁²_𝑙(𝛼,𝜙) in Theorem 11.(4)The index sequence {𝑃_𝑖} is admissible and regulated.Then the sequence {𝑥^(𝑘)} generated by Algorithm 9 converges for any initial 𝑥⁽⁰⁾ if and only if (𝛼,𝛽,𝛽_𝑖,𝛾_𝑖,𝜔), 𝑖=1,2,…𝑛∈𝑊₃, with 𝜌=(|𝐵|) and 𝜃=max_{1≤𝑙≤𝑠}{‖𝑃₁⁻¹(𝑀¹_𝑙)⁻¹(𝛽,𝜓)𝑁¹_𝑙(𝛽,𝜓)𝑃₁‖_∞‖𝑃₂⁻¹(𝑀²_𝑙)⁻¹ (𝛼,𝜙)𝑁²_𝑙(𝛼,𝜙)𝑃₂‖_∞}, where 𝑊₃=𝛼,𝛽,𝛽_𝑖,𝛾_𝑖,𝜔∈𝑅⁵∶0≤𝛼,𝛽≤1,0<𝛽_𝑖,𝛾_𝑖<221+𝜌,0<𝜔<.1+𝜃,𝑖=1,2,…,𝑛. (33)

Remark 14. It is known that an 𝑀-matrix or a symmetric positive definite 𝐿-matrix is also 𝐻-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, ‖𝐵‖_∞ can take the place of 𝜌 in Theorems 11, 12, and 13.

4. Applied Convergence of Relaxed Chaotic Methods

In Theorems 12 and 13 of this paper, we find that 𝜃 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 𝐴∈𝑅^𝑛×𝑛 be an irreducible diagonally dominant matrix and (𝐼−𝐿_𝑙,𝑈_𝑙,𝐸_𝑙), 𝑙=1,2,…,𝑠, a multisplitting of 𝐴. Assume that for 𝑙=1,2,…,𝑠, we have the following.(1)𝐿_𝑙 are the strictly lower triangular matrices and 𝑈_𝑙 are the matrices such that the equalities 𝐴=𝐼−𝐿_𝑙−𝑈_𝑙 hold.(2)⟨𝐴⟩=|𝐼|−|𝐿_𝑙|−|𝑈_𝑙|=|𝐼|−|𝐵|, where |𝐵|=|𝐿_𝑙|+|𝑈_𝑙|.(3)𝑃 is diagonal matrix defined in Lemma 3 and 𝑀¹_𝑙(𝛽,𝜓), 𝑁¹_𝑙(𝛽,𝜓), 𝑀²_𝑙(𝛼,𝜙), and 𝑁²_𝑙(𝛼,𝜙) in Theorem 11.Then the sequence {𝑥^(𝑘)} generated by Algorithm 7 converges for any initial 𝑥⁽⁰⁾ if and only if (𝛼,𝛽,𝛽_𝑖,𝛾_𝑖,𝜔),  𝑖=1,2,…,𝑛∈𝑊₄, with 𝜃^′=max_{1≤𝑙≤𝑛}{(|1−𝛾_𝑖|+𝛾_𝑖‖𝐵‖_∞)}max_{1≤𝑙≤𝑛}{(|1−𝛽_𝑖|+𝛽_𝑖‖𝐵‖_∞)}, where 𝑊₄=𝛼,𝛽,𝛽_𝑖,𝛾_𝑖,𝜔∈𝑅⁵∶0≤𝛼,𝛽≤1,0<𝛽_𝑖,𝛾_𝑖<21+‖𝐵‖_∞2,0<𝜔<1+𝜃^′. (34)

Proof. We only prove |(𝐻_GUSAOR)_𝑘|𝑥<𝜃^′′𝑥(𝑘=1,2,…,0≤𝜃^′≤1). Let us define the iteration matrix in Algorithm 7𝐻_GUSAOR𝑘=𝜔𝐻_GUSAOR𝑘+(1−𝜔)𝐼. (35) From the proving process of Theorem 11, we may define 𝑇₁=𝐼−𝛽𝜓𝑈_𝑙⁻¹𝐼−𝜓+1−𝛽𝜓𝑈_𝑙+𝜓𝐿_𝑙,𝑇₂=𝐼−𝛼𝜙𝐿_𝑙⁻¹𝐼−𝜙+(1−𝛼)𝜙𝐿_𝑙+𝜙𝑈_𝑙. (36) At first, we need to prove 𝜌(𝑇₁)<1. Similarly, we may also prove 𝜌(𝑇₂)<1. Assume that 0≤𝛼,𝛽≤1,0<𝛽_𝑖,𝛾_𝑖<2.1+𝜌 (37) With (36), we know that 𝑇₁ is nonnegative, so according to [24, Theorem 2.7], there exists an eigenvector 𝑥≥0, 𝑥≠0, such that 𝑇₁𝑥=𝜌(𝑇₁)𝑥 hold; that is, 𝐼−𝜓+1−𝛽𝜓𝑈_𝑙+𝜓𝐿_𝑙𝑥=𝜌𝑇₁𝐼−𝛽𝜓𝑈_𝑙𝑥. (38) Multiplying by 𝜓⁻¹, it holds that 𝜌𝑇₁𝜓⁻¹−𝐼−𝜓⁻¹𝑥=1−𝛽+𝛽𝜌𝑇₁𝑈_𝑙+𝐿_𝑙𝑥. (39) As [1−𝛽+𝛽𝜌(𝑇₁)]|𝑈_𝑙|+|𝐿_𝑙|≥0, it follows by [5, Theorem 11] that min_{1≤𝑙≤𝑛}𝛾_𝑖⁻¹𝜌𝑇₁−1−𝛾_𝑖⁻¹≤𝜌1−𝛽+𝛽𝜌𝑇₁𝑈_𝑙+𝐿_𝑙. (40) From the proof of [3, Theorem 3.1], we have 𝜌(𝑇₁)<1, 0≤1−𝛽+𝛽𝜌(𝑇₁)<1. Since the matrices 𝐴, 𝐵, and [1−𝛽+𝛽𝜌(𝑇₁)]|𝑈_𝑙|+|𝐿_𝑙| are irreducible, by [24, Theorem 2.1], it follows that 𝜌1−𝛽+𝛽𝜌𝑇₁𝑈_𝑙+𝐿_𝑙<𝜌𝐿_𝑙+𝑈_𝑙=𝜌(|𝐵|)≤‖𝐵‖_∞. (41) With (40) and (41) and 𝑀 being irreducible diagonally dominant by rows, we have 𝜌𝑇₁<max_{1≤𝑙≤𝑛}1−𝛾_𝑖+𝛾_𝑖‖𝐵‖_∞. (42) Notice that if 𝛾_𝑖≤1, we may obtain 1−𝛾_𝑖+𝛾_𝑖‖𝐵‖_∞=1−𝛾_𝑖+𝛾_𝑖‖𝐵‖_∞<1. (43) While if 1<𝛾_𝑖<2/(1+‖𝐵‖_∞), we also have 1−𝛾_𝑖+𝛾_𝑖‖𝐵‖_∞=−1+𝛾_𝑖+𝛾_𝑖‖𝐵‖_∞=𝛾_𝑖1+‖𝐵‖_∞<2−11+‖𝐵‖_∞1+‖𝐵‖_∞−1=1, (44) From (42), (43), and (44), we can get 𝜌𝑇₁<max_{1≤𝑙≤𝑛}1−𝛾_𝑖+𝛾_𝑖‖𝐵‖_∞𝜌𝑇<1,₂<max_{1≤𝑙≤𝑛}1−𝛽_𝑖+𝛽_𝑖‖𝐵‖_∞<1. (45) From Theorem 11, we have 𝐻_GUSAOR𝑘≤𝐻_GUSAOR𝑘=^𝑠_𝑙=1𝐻𝛽,𝜓_𝑙𝐻𝛼,𝜙_𝑙≤^𝑠_𝑙=1𝐻𝛽,𝜓_𝑙𝐻𝛼,𝜙_𝑙≤^𝑠_𝑙=1𝑇₁𝑇₂. (46) From (35) and the above proof, we have 𝐻_GUSAOR𝑘𝑥≤𝜔𝐻_GUSAOR𝑘𝑥+|1−𝜔|𝑥≤𝜔^𝑠_𝑙=1𝐸_𝑙𝐻𝛽,𝜓_𝑙𝐻𝛼,𝜙_𝑙𝑥+|1−𝜔|𝑥≤𝜔^𝑠_𝑙=1𝐸_𝑙𝐻𝛽,𝜓_𝑙𝐻𝛼,𝜙_𝑙||𝑥≤𝑥+1−𝜔_𝑠𝑙=1𝐸_𝑙𝑇₁𝑇₂||𝑥𝑇𝑥+1−𝜔=𝜌₂𝜔^𝑠_𝑙=1𝐸_𝑙𝑇₁||𝑥𝑇𝑥+1−𝜔=𝜔𝜌₁𝜌𝑇₂^𝑠_𝑙=1𝐸_𝑙||𝑥𝑇𝑥+1−𝜔=𝜔𝜌₁𝜌𝑇₂𝑥+|1−𝜔|𝑥<𝜔max_{1≤𝑙≤𝑛}1−𝛾_𝑖+𝛾_𝑖‖𝐵‖_∞×max_{1≤𝑙≤𝑛}1−𝛽_𝑖+𝛽_𝑖‖𝐵‖_∞||𝑥=𝑥+1−𝜔𝜔𝜃^′+|1−𝜔|𝑥=𝜃^′′𝑥, (47) where 𝜃^′=max_{1≤𝑙≤𝑛}{|1−𝛾_𝑖|+𝛾_𝑖‖𝐵‖_∞}max_{1≤𝑙≤𝑛}{|1−𝛽_𝑖|+𝛽_𝑖‖𝐵‖_∞}, 0<𝜔<2/(1+𝜃^′), 𝜃^′′=𝜔𝜃^′+|1−𝜔|<1.

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 𝐴∈𝑅^𝑛×𝑛 be an irreducible diagonally dominant matrix and (𝐼−𝐿_𝑙,𝑈_𝑙,𝐸_𝑙), 𝑙=1,2,…,𝑠, a multisplitting of 𝐴. Assume that for 𝑙=1,2,…,𝑠, we have the following.(1)𝐿_𝑙 are the strictly lower triangular matrices and 𝑈_𝑙 are the matrices such that the equalities 𝐴=𝐼−𝐿_𝑙−𝑈_𝑙 hold.(2)⟨𝐴⟩=|𝐼|−|𝐿_𝑙|−|𝑈_𝑙|=|𝐼|−|𝐵|, where |𝐵|=|𝐿_𝑙|+|𝑈_𝑙|.(3)𝑃 is diagonal matrix defined in Lemma 3 and 𝑀¹_𝑙(𝛽,𝜓), 𝑁¹_𝑙(𝛽,𝜓), 𝑀²_𝑙(𝛼,𝜙), and 𝑁²_𝑙(𝛼,𝜙) in Theorem 11.(4)The index sequence {𝑃_𝑖} is admissible and regulated.Then the sequence {𝑥^(𝑘)} generated by Algorithm 9 converges for any initial 𝑥⁽⁰⁾ if and only if (𝛼,𝛽,𝛽_𝑖,𝛾_𝑖,𝜔),  𝑖=1,2,…,𝑛∈𝑊₅, with 𝜃^′=max_{1≤𝑙≤𝑛}{(|1−𝛾_𝑖|+𝛾_𝑖‖𝐵‖_∞)}max_{1≤𝑙≤𝑛}{(|1−𝛽_𝑖|+𝛽_𝑖‖𝐵‖_∞)}, where 𝑊₅=𝛼,𝛽,𝛽_𝑖,𝛾_𝑖,𝜔∈𝑅⁵∶0≤𝛼,𝛽≤1,0<𝛽_𝑖,𝛾_𝑖<21+‖𝐵‖_∞2,0<𝜔<1+𝜃^′. (48)

Remark 19. As a special case, for the relaxed matrix multisplitting chaotic GSSOR-style methods, we have the corresponding convergence results, where 𝛼=1, 𝛽=1, 𝜙=diag(𝛽₁,𝛽₂,…,𝛽_𝑛), 𝜓=diag(𝛾₁,𝛾₂,…,𝛾_𝑛) with 𝛽_𝑖≥0, 𝛾_𝑖≥0(𝑖=1,2,…,𝑛) and 𝜔 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 (𝑛=6), which is as follows: 1𝐴=−14300001140100031−13300004111000041−143000011,𝑏=4114311412. (49)

Now, we will apply the results of Theorem 16 according to Algorithm 7. From Algorithm 7, we can get the iterative matrix 𝐻=𝜔^𝑠_𝑙=1𝐸_𝑙𝐻𝛽,𝜓_𝑙𝐻𝛼,𝜙_𝑙+(1−𝜔). (50) Here, we assume that 𝑠=3, 𝜇_𝑙,𝑘≡1. By direct calculations with Matlab 7.1, we have ‖𝐵‖_∞2=1.75,1+‖𝐵‖_∞=0.7273. (51) In Table 1, we show that convenience results of Section 4 are convenience for carrying out numerical experiments, where 𝜌(𝐻) denote the spectral radius of iterative matrix 𝐻: 𝐸₁=diag(1,1,0,0,0,0),𝐸₂(),𝐸=diag0,0,1,1,0,0₃().𝐿=diag0,0,0,0,1,1₁=100000020100000330000001200000004,𝐿000000−0.40₂=−300000040100000670000001200000008,𝐿0000000.20₃=30000000.30000000.5000000−0.3000000,𝑈400000000.40_𝑙=𝐼−𝐿_𝑙−𝐴,𝑙=1,2,3. (52)

Table 1 

Convergence results of Theorems 16 and 18.

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

Example 2. Consider a matrix 𝐴∈𝑅^𝑛×𝑛 of the form =−,𝐴=100⋯0−𝜈−𝜈10⋯000−𝜈1⋯00⋮⋮⋱⋮⋮⋮000⋯10000⋯−𝜈1100⋯00010⋯00001⋯00⋮⋮⋱⋮⋮⋮000⋯10000⋯01000⋯0𝜈𝜈00⋯000𝜈0⋯00⋮⋮⋱⋮⋮⋮000⋯00000⋯𝜈0 (53) where 𝜌(|𝐷|⁻¹|𝐵|)=𝜈, 𝜈 is a real number, satisfying |𝜈|<1.

In Theorem 11, let 0≤𝛼, 𝛽≤1, 0<𝛽_𝑖, 𝛾_𝑖<2/(1+𝜈), 𝑖=1,2,…,𝑛; then Algorithm 5 converges for any initial vector 𝑥⁽⁰⁾. In Theorems 12 and 13, let 0≤𝛼, 𝛽≤1, 0<𝛽_𝑖, 𝛾_𝑖<2/(1+𝜈), 0<𝜔<2/(1+𝜃), 𝑖=1,2,…,𝑛; then Algorithms 7 and 9 converge for any initial vector 𝑥⁽⁰⁾. In Theorems 16 and 18, let 0≤𝛼, 𝛽≤1, 0<𝛽_𝑖, 𝛾_𝑖<2/(1+𝜈), 0<𝜔<2/(1+𝜃^′), 𝑖=1,2,…,𝑛; then Algorithms 7 and 9 converge for any initial vector 𝑥⁽⁰⁾, where 𝜃^′=max_{1≤𝑙≤𝑛}{(|1−𝛾_𝑖|+𝜈𝛾_𝑖)}max_{1≤𝑙≤𝑛}{(|1−𝛽_𝑖|+𝜈𝛽_𝑖)}.

If we choose 𝜈=1/5, in Theorem 11, let 0≤𝛼, 𝛽≤1, 0<𝛽_𝑖, 𝛾_𝑖<5/3, 𝑖=1,2,…,𝑛; then Algorithm 5 converges for any initial vector 𝑥⁽⁰⁾. In Theorems 12 and 13, let 0≤𝛼, 𝛽≤1, 0<𝛽_𝑖, 𝛾_𝑖<5/3, 0<𝜔<2/(1+𝜃), 𝑖=1,2,…,𝑛; then Algorithms 7 and 9 converge for any initial vector 𝑥⁽⁰⁾. In Theorems 16 and 18, let 0≤𝛼, 𝛽≤1, 0<𝛽_𝑖, 𝛾_𝑖<5/3, 0<𝜔<2/(1+𝜃^′), 𝑖=1,2,…,𝑛; then Algorithms 7 and 9 converge for any initial vector 𝑥⁽⁰⁾, where 𝜃^′=max_{1≤𝑙≤𝑛}{(|1−𝛾_𝑖|+(1/5)𝛾_𝑖)}max_{1≤𝑙≤𝑛}{(|1−𝛽_𝑖|+(1/5)𝛽_𝑖)}.

6. Conclusions

In this paper, we consider relaxed matrix parallel multisplitting chaotic GUSAOR-style methods for solving linear systems of algebraic equations 𝐴𝑥=𝑏, in which the coefficient 𝐴 is an 𝐻-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).