Improving Results on Convergence of AOR Method

Abstract

We present some sufficient conditions on convergence of AOR method for solving 𝐴𝑥=𝑏 with 𝐴 being a strictly doubly 𝛼 diagonally dominant matrix. Moreover, we give two numerical examples to show the advantage of the new results.

1. Introduction

Let us denote all complex square matrices by 𝐶^𝑛×𝑛 and all complex vectors by 𝐶^𝑛.

For 𝐴=(𝑎_𝑖𝑗)∈𝐶^𝑛×𝑛, we denote by 𝜌(𝐴) the spectral radius of matrix 𝐴.

Let us consider linear system 𝐴𝑥=𝑏, where 𝑏∈𝐶^𝑛 is a given vector and 𝑥∈𝐶^𝑛 is an unknown vector. Let 𝐴=𝐷−𝑇−𝑆 be given and 𝐷 is the diagonal matrix, −𝑇 and −𝑆 are strictly lower and strictly upper triangular parts of 𝐴, respectively, and denote 𝐿=𝐷⁻¹𝑇,𝑈=𝐷⁻¹𝑆, (1.1) where det(𝐷)≠0.

Then the AOR method [1] can be written as 𝑥^𝑘+1=𝑀_𝜎,𝜔𝑥^𝑘+𝑑,𝑘=0,1,…,𝑥⁰∈𝐶^𝑛, (1.2) where 𝑀_𝜎,𝜔=(𝐼−𝜎𝐿)⁻¹)()],()[(1−𝜔𝐼+𝜔−𝜎𝐿+𝜔𝑈𝑑=𝜔𝐼−𝜎𝐿⁻¹𝑏,𝜔,𝜎∈𝑅. (1.3)

2. Preliminaries

We denote 𝑅_𝑖(𝐴)=_𝑗≠𝑖𝑎_𝑖𝑗,𝑆_𝑖(𝐴)=_𝑗≠𝑖𝑎_𝑗𝑖,𝑃_𝑖,𝛼(𝐴)=𝛼𝑅_𝑖(𝐴)+(1−𝛼)𝑆_𝑖(𝐴),∀𝑖∈𝑁. (2.1) For any matrix 𝐴=(𝑎_𝑖𝑗)∈𝐶^𝑛×𝑛, the comparison matrix 𝑀(𝐴)=(𝑚_𝑖𝑗)∈𝑅^𝑛×𝑛 is defined by 𝑚_𝑖𝑖=𝑎_𝑖𝑖,𝑚_𝑖𝑗=−𝑎_𝑖𝑗,𝑖,𝑗∈𝑁,𝑖≠𝑗,𝑁={1,2,…,𝑛}. (2.2)

Definition 2.1 (see [2]). A matrix 𝐴∈𝐶^𝑛×𝑛 is called a strictly diagonally dominant matrix (𝑆𝐷) if 𝑎_𝑖𝑖>𝑅_𝑖(𝐴),∀𝑖∈𝑁. (2.3) A matrix 𝐴∈𝐶^𝑛×𝑛 is called a strictly doubly diagonally dominant matrix (𝐷𝐷) if 𝑎_𝑖𝑖𝑎_𝑗𝑗>𝑅_𝑖(𝐴)𝑅_𝑗(𝐴),∀𝑖,𝑗∈𝑁,𝑖≠𝑗. (2.4)

Definition 2.2 (see [3]). A matrix 𝐴∈𝐶^𝑛×𝑛 is called a strictly 𝛼 diagonally dominant matrix (𝐷(𝛼)) if there exits 𝛼∈[0,1], such that 𝑎_𝑖𝑖>𝛼𝑅_𝑖(𝐴)+(1−𝛼)𝑆_𝑖(𝐴),∀𝑖∈𝑁. (2.5)

Definition 2.3 (see [4]). Let 𝐴=(𝑎_𝑖𝑗)∈𝐶^𝑛×𝑛, if there exits 𝛼∈[0,1] such that 𝑎_𝑖𝑖𝑎_𝑗𝑗>𝛼𝑅_𝑖(𝐴)𝑅_𝑗(𝐴)+(1−𝛼)𝑆_𝑖(𝐴)𝑆_𝑗(𝐴),∀𝑖,𝑗∈𝑁,𝑖≠𝑗, (2.6) then 𝐴 is called a strictly doubly 𝛼 diagonally dominant matrix (𝐷𝐷(𝛼)).

In [3, 5, 6], some people studied the convergence of AOR method for solving linear system 𝐴𝑥=𝑏 and gave the areas of convergence. In [5], Cvetković and Herceg studied the convergence of AOR method for strictly diagonally dominant matrices. In [3], Huang and Wang studied the convergence of AOR method for strictly 𝛼 diagonally dominant matrices. In [6], Gao and Huang studied the convergence of AOR method for strictly doubly diagonally dominant matrices.

Theorem 2.4 (see [3]). Let 𝐴∈𝐷(𝛼), then AOR method converges for (I)20≤𝜎<1+𝜌𝑀_0,1(𝑀(𝐴2))=𝑠,0<𝜔<max1+max_𝑖𝑃_𝑖,𝛼(𝐿+𝑈)=𝑡,2𝜎1+𝜌𝑀_𝜎,𝜎(),𝑜𝑟IImax_𝑖−𝜔1−𝑃_𝑖,𝛼(𝐿+𝑈)+2max(0,𝜔−1)2𝑃_𝑖,𝛼(𝐿)()<𝜎<0,0<𝜔<𝑡,𝑜𝑟III𝑡≤𝜎<min_𝑖𝜔1+𝑃_𝑖,𝛼(𝐿)−𝑃_𝑖,𝛼(𝑈)+2min(0,1−𝜔)2𝑃_𝑖,𝛼(𝐿),0<𝜔<𝑡. (2.7)

Theorem 2.5 (see [6]). Let 𝐴∈𝐷𝐷, then AOR method converges for (I)20≤𝜎<1+𝜌𝑀_0,1(𝑀(𝐴))=𝑠,0<𝜔<max2𝜎1+𝜌𝑀_𝜎,𝜎,min_{𝑖,𝑗𝑖≠𝑗}21+𝑅_𝑖(𝐿+𝑈)𝑅_𝑗()()𝐿+𝑈=𝑡,𝑜𝑟IImax_{𝑖,𝑗𝑖≠𝑗}𝜔𝑃₁−𝜔²𝑃²₂+𝑃₃min𝜔²,(𝜔−2)²𝑃₃()<𝜎<0,0<𝜔<𝑡,𝑜𝑟III𝑡≤𝜎<min_{𝑖,𝑗𝑖≠𝑗}𝜔𝑃₄+𝜔²𝑃²₅+𝑃₃min𝜔²,(𝜔−2)²𝑃₃,0<𝜔<𝑡, (2.8) where 𝑃₁=𝑅_𝑖(𝐿)𝑅_𝑗(𝐿+𝑈)+𝑅_𝑖(𝐿+𝑈)𝑅_𝑗(𝐿),𝑃₂=𝑅_𝑖(𝐿)𝑅_𝑗(𝐿+𝑈)−𝑅_𝑖(𝐿+𝑈)𝑅_𝑗(𝐿),𝑃₃=4𝑅_𝑖(𝐿)𝑅_𝑗(𝐿),𝑃₄=𝑅_𝑖(𝐿)𝑅_𝑗(𝐿)−𝑅_𝑗(𝑈)+𝑅_𝑗(𝐿)𝑅_𝑖(𝐿)−𝑅_𝑖(𝑈),𝑃₅=𝑅_𝑖(𝐿)𝑅_𝑗(𝐿)−𝑅_𝑗(𝑈)−𝑅_𝑗(𝐿)𝑅_𝑖(𝐿)−𝑅_𝑖(𝑈). (2.9)

3. Upper Bound for Spectral Radius of 𝑀_𝜎,𝜔

In the following, we present an upper bound for spectral radius of AOR iterative matrix 𝑀_𝜎,𝜔 for strictly doubly 𝛼 diagonally dominant coefficient matrix.

Lemma 3.1 (see [4]). If 𝐴∈𝐷𝐷(𝛼), then 𝐴 is a nonsingular H-matrix.

Theorem 3.2. Let 𝐴∈𝐷𝐷(𝛼), if 1−𝜎²[𝛼𝑅_𝑖(𝐿)𝑅_𝑗(𝐿)+(1−𝛼)𝑆_𝑖(𝐿)𝑆_𝑗(𝐿)]>0, for  all  𝑗∈𝑁,𝑖≠𝑗, then 𝜌𝑀_𝜎,𝜔≤max_{𝑖,𝑗𝑖≠𝑗}𝐴₂+𝐴²₂−4𝐴₁𝐴₃2𝐴₁, (3.1) where 𝐴₁=1−𝜎²𝛼𝑅_𝑖(𝐿)𝑅_𝑗(𝐿)+(1−𝛼)𝑆_𝑖(𝐿)𝑆_𝑗(𝐿,𝐴₂)=2|1−𝜔|+2|𝜔−𝜎||𝜎|𝛼𝑅_𝑖(𝐿)𝑅_𝑗(𝐿)+(1−𝛼)𝑆_𝑖(𝐿)𝑆_𝑗(𝐿+|𝜔𝜎)|||𝛼𝑅_𝑖(𝐿)𝑅_𝑗(𝑈)+(1−𝛼)𝑆_𝑖(𝑈)𝑆_𝑗(𝐿+|𝜔|𝜎)||𝛼𝑅_𝑖(𝑈)𝑅_𝑗(𝐿)+(1−𝛼)𝑆_𝑖(𝐿)𝑆_𝑗(𝑈,𝐴₃)=(1−𝜔)²−𝛼|𝜔−𝜎|𝑅_𝑖(𝐿)+|𝜔|𝑅_𝑖(𝑈)|𝜔−𝜎|𝑅_𝑗(𝐿)+|𝜔|𝑅_𝑗(𝑈−(|𝑆)1−𝛼)|𝜔−𝜎_𝑖(𝐿)+|𝜔|𝑆_𝑖(𝑈)|𝜔−𝜎|𝑆_𝑗(𝐿)+|𝜔|𝑆_𝑗(𝑈). (3.2)

Proof. Let 𝜆 be an eigenvalue of 𝑀_𝜎,𝜔 such that det𝜆𝐼−𝑀_𝜎,𝜔=0, (3.3) that is, det(𝜆(𝐼−𝜎𝐿)−((1−𝜔)𝐼+(𝜔−𝜎)𝐿+𝜔𝑈))=0. (3.4) If 𝜆(𝐼−𝜎𝐿)−((1−𝜔)𝐼+(𝜔−𝜎)𝐿+𝜔𝑈)∈𝐷𝐷(𝛼), then by Lemma 3.1, 𝜆(𝐼−𝜎𝐿)−((1−𝜔)𝐼+(𝜔−𝜎)𝐿+𝜔𝑈) is nonsingular and 𝜆 is not an eigenvalue of iterative matrix 𝑀_𝜎,𝜔, that is, if |𝜆−(1−𝜔)|²>𝛼_𝑡≠𝑖𝜆𝜎𝑙_𝑖𝑡−(𝜔−𝜎)𝑙_𝑖𝑡−𝜔𝑢_{𝑖𝑡𝑙≠𝑗}𝜆𝜎𝑙_𝑗𝑙−(𝜔−𝜎)𝑙_𝑗𝑙−𝜔𝑢_𝑗𝑙+(1−𝛼)_𝑘≠𝑖𝜆𝜎𝑙_𝑘𝑖−(𝜔−𝜎)𝑙_𝑘𝑖−𝜔𝑢_𝑘𝑖×_𝑠≠𝑗𝜆𝜎𝑙_𝑠𝑗−(𝜔−𝜎)𝑙_𝑠𝑗−𝜔𝑢_𝑠𝑗,∀𝑖,𝑗∈𝑁,𝑖≠𝑗, (3.5) then 𝜆 is not an eigenvalue of 𝑀_𝜎,𝜔. Especially, if (|𝜆|−|1−𝜔|)²>𝛼_𝑡≠𝑖|𝜆||𝜎|𝑙_𝑖𝑡+|𝜔−𝜎|𝑙_𝑖𝑡+|𝜔|𝑢_𝑖𝑡×_𝑙≠𝑗|𝜆||𝜎|𝑙_𝑗𝑙+|𝜔−𝜎|𝑙_𝑗𝑙+|𝜔|𝑢_𝑗𝑙+(1−𝛼)_𝑘≠𝑖|𝜆||𝜎|𝑙_𝑘𝑖+|𝜔−𝜎|𝑙_𝑘𝑖+|𝜔|𝑢_𝑘𝑖×_𝑠≠𝑗|𝜆||𝜎|𝑙_𝑠𝑗+|𝜔−𝜎|𝑙_𝑠𝑗+|𝜔|𝑢_𝑠𝑗, (3.6) then 𝜆 is not an eigenvalue of 𝑀_𝜎,𝜔.
If 𝜆 is an eigenvalue of 𝑀_𝜎,𝜔, then there exits at least a couple of 𝑖,𝑗(𝑖≠𝑗), such that (|𝜆|−|1−𝜔|)²≤𝛼_𝑡≠𝑖|𝜆||𝜎|𝑙_𝑖𝑡+|𝜔−𝜎|𝑙_𝑖𝑡+|𝜔|𝑢_𝑖𝑡×_𝑙≠𝑗|𝜆||𝜎|𝑙_𝑗𝑙+|𝜔−𝜎|𝑙_𝑗𝑙+|𝜔|𝑢_𝑗𝑙+(1−𝛼)_𝑘≠𝑖|𝜆||𝜎|𝑙_𝑘𝑖+|𝜔−𝜎|𝑙_𝑘𝑖+|𝜔|𝑢_𝑘𝑖×_𝑠≠𝑗|𝜆||𝜎|𝑙_𝑠𝑗+|𝜔−𝜎|𝑙_𝑠𝑗+|𝜔|𝑢_𝑠𝑗, (3.7) that is, 𝐴₁|𝜆|²−𝐴₂|𝜆|+𝐴₃≤0. (3.8) Since 𝐴₁=1−𝜎²[𝛼𝑅_𝑖(𝐿)𝑅_𝑗(𝐿)+(1−𝛼)𝑆_𝑖(𝐿)𝑆_𝑗(𝐿)]>0, and the discriminant Δ of the quadratic in |𝜆| satisfies Δ≥0, then the solution of (3.8) satisfies 𝐴₂−𝐴²₂−4𝐴₁𝐴₃2𝐴₁≤|𝜆|≤𝐴₂+𝐴²₂−4𝐴₁𝐴₃2𝐴₁. (3.9) So 𝜌𝑀_𝜎,𝜔≤max_{𝑖,𝑗𝑖≠𝑗}𝐴₂+𝐴²₂−4𝐴₁𝐴₃2𝐴₁. (3.10)

4. Improving Results on Convergence of AOR Method

In this section, we present new results on convergence of AOR method.

Theorem 4.1. Let 𝐴∈𝐷𝐷(𝛼), then AOR method converges if 𝜔,𝜎 satisfy either (I)20≤𝜎<1+𝜌𝑀_0,1(𝑀(𝐴))=𝑠,0<𝜔<max2𝜎1+𝜌𝑀_𝜎,𝜎,min_{𝑖,𝑗𝑖≠𝑗}21+𝛼𝑅_𝑖(𝐿+𝑈)𝑅_𝑗(𝐿+𝑈)+(1−𝛼)𝑆_𝑖(𝐿+𝑈)𝑆_𝑗()()𝐿+𝑈=𝑡,𝑜𝑟IImax_{𝑖,𝑗𝑖≠𝑗}𝜔𝑃^′₁−𝜔²𝑃^′2₂+𝑃^′₃min𝜔²,(𝜔−2)²𝑃^′₃()<𝜎<0,0<𝜔<𝑡,𝑜𝑟III𝑡≤𝜎<min_{𝑖,𝑗𝑖≠𝑗}𝜔𝑃^′₄+𝜔²𝑃^′2₅+𝑃^′₃min𝜔²,(𝜔−2)²𝑃^′₃,0<𝜔<𝑡, (4.1) where 𝑃^′₁=𝛼𝑅_𝑖(𝐿)𝑅_𝑗(𝐿+𝑈)+(1−𝛼)𝑆_𝑖(𝐿)𝑆_𝑗(𝐿+𝑈)+𝛼𝑅_𝑖(𝐿+𝑈)𝑅_𝑗(𝐿)+(1−𝛼)𝑆_𝑖(𝐿+𝑈)𝑆_𝑗(𝐿),𝑃^′₂=𝛼𝑅_𝑖(𝐿)𝑅_𝑗(𝐿+𝑈)+(1−𝛼)𝑆_𝑖(𝐿)𝑆_𝑗(𝐿+𝑈)−𝛼𝑅_𝑖(𝐿+𝑈)𝑅_𝑗(𝐿)−(1−𝛼)𝑆_𝑖(𝐿+𝑈)𝑆_𝑗(𝐿),𝑃^′₃=4𝛼𝑅_𝑖(𝐿)𝑅_𝑗(𝐿)+(1−𝛼)𝑆_𝑖(𝐿)𝑆_𝑗(𝐿,𝑃_′4)=𝛼𝑅_𝑖(𝐿)𝑅_𝑗(𝐿)−𝑅_𝑗(𝑈)+(1−𝛼)𝑆_𝑖(𝐿)𝑆_𝑗(𝐿)−𝑆_𝑗(𝑈)+𝛼𝑅_𝑗(𝐿)𝑅_𝑖(𝐿)−𝑅_𝑖(𝑈)+(1−𝛼)𝑆_𝑗(𝐿)𝑆_𝑖(𝐿)−𝑆_𝑖(𝑈,𝑃_′5)=𝛼𝑅_𝑖(𝐿)𝑅_𝑗(𝐿)−𝑅_𝑗(𝑈)+(1−𝛼)𝑆_𝑖(𝐿)𝑆_𝑗(𝐿)−𝑆_𝑗(𝑈)−𝛼𝑅_𝑗(𝐿)𝑅_𝑖(𝐿)−𝑅_𝑖(𝑈)−(1−𝛼)𝑆_𝑗(𝐿)𝑆_𝑖(𝐿)−𝑆_𝑖(𝑈). (4.2)

Proof. It is easy to verify that for each 𝜎, which satisfies one of the conditions (I)–(III), we have 1−𝜎²𝛼𝑅_𝑖(𝐿)𝑅_𝑗(𝐿)+(1−𝛼)𝑆_𝑖(𝐿)𝑆_𝑗(𝐿)>0,∀𝑖≠𝑗;𝑖,𝑗∈𝑁. (4.3)
Firstly, we consider case (I). Since 𝐴 be a 𝛼 diagonally dominant matrix, then by Lemma 3.1, we know that 𝐴 is a nonsingular H-matrix; therefore, 𝑀(𝐴) is a nonsingular M-matrix, and it follows that from paper [7], 𝜌(𝑀_𝜎,𝜎)<1 holds for 0≤𝜎<𝑠 and for 𝜎≠0, 𝑀_𝜎,𝜔=𝜔1−𝜎𝜔𝐼+𝜎𝑀_𝜎,𝜎. (4.4) If 0<𝜔<max{2𝜎/(1+𝜌(𝑀_𝜎,𝜎)),𝑡}=2𝜎/(1+𝜌(𝑀_𝜎,𝜎)), then 𝜔0<𝜎<21+𝜌𝑀_𝜎,𝜎, (4.5) by extrapolation theorem [6], we have 𝜌(𝑀_𝜎,𝜔)<1.
If 0<𝜔<max{2𝜎/(1+𝜌(𝑀_𝜎,𝜎)),𝑡}=𝑡, then it remains to analyze the case 2𝜎1+𝜌𝑀_𝜎,𝜎≤𝜔<𝑡,0≤𝜎<𝑠. (4.6) Since when 𝜌(𝑀_𝜎,𝜎)<1, 𝜎<2𝜎1+𝜌𝑀_𝜎,𝜎, (4.7) then 0≤𝜎<𝜔. From 𝐴₂−𝐴₃<𝐴₁(a)𝐴₂<2𝐴₁(b)⟺𝐴≤2²+𝐴²₂−4𝐴₁𝐴₃2𝐴₁<1,𝐴²₂−4𝐴₁𝐴₃<2𝐴₁≤2(c) (4.8) we have 𝜌(𝑀_𝜎,𝜔)≤max_{𝑖,𝑗𝑖≠𝑗}((𝐴₂+𝐴²₂−4𝐴₁𝐴₃)/2𝐴₁)<1.(1)When 𝜔≤1, it easy to verify that (4.8) holds. (2)When 𝜔>1, since 𝐴₁=1−𝜎²𝛼𝑅_𝑖(𝐿)𝑅_𝑗(𝐿)+(1−𝛼)𝑆_𝑖(𝐿)𝑆_𝑗(𝐿,𝐴₂)=2(𝜔−𝜎)𝛼𝑅_𝑖(𝐿)𝑅_𝑗(𝐿)+(1−𝛼)𝑆_𝑖(𝐿)𝑆_𝑗(𝐿)+𝜔𝜎𝛼𝑅_𝑖(𝐿)𝑅_𝑗(𝑈)+(1−𝛼)𝑆_𝑖(𝑈)𝑆_𝑗(𝐿)+𝜔𝜎𝛼𝑅_𝑖(𝑈)𝑅_𝑗(𝐿)+(1−𝛼)𝑆_𝑖(𝐿)𝑆_𝑗(𝑈||,𝐴)+21−𝜔₃=(1−𝜔)²−𝛼(𝜔−𝜎)𝑅_𝑖(𝐿)+𝜔𝑅_𝑖(𝑈)(𝜔−𝜎)𝑅_𝑗(𝐿)+𝜔𝑅_𝑗(𝑈−()𝑆)1−𝛼)(𝜔−𝜎_𝑖(𝐿)+𝜔𝑆_𝑖(𝑈)(𝜔−𝜎)𝑆_𝑗(𝐿)+𝜔𝑆_𝑗(𝑈), (4.9)then by 𝐴₂−𝐴₃<𝐴₁ and 1−𝜎²[𝛼𝑅_𝑖(𝐿)𝑅_𝑗(𝐿)+(1−𝛼)𝑆_𝑖(𝐿)𝑆_𝑗(𝐿)]>0, forall𝑖,𝑗∈𝑁,𝑖≠𝑗, we have 𝜔²1−𝛼𝑅_𝑖(𝐿+𝑈)𝑅_𝑗(𝐿+𝑈)−(1−𝛼)𝑆_𝑖(𝐿+𝑈)𝑆_𝑗(𝐿+𝑈)−4𝜔+4>0. (4.10) It is easy to verify that the discriminant Δ of the quadratic in 𝜔 satisfies Δ>0, and so there holds 𝜔₁>21−𝛼𝑅_𝑖(𝐿+𝑈)𝑅_𝑗(𝐿+𝑈)−(1−𝛼)𝑆_𝑖(𝐿+𝑈)𝑆_𝑗(),𝐿+𝑈 (4.11) or 𝜔₂<21+𝛼𝑅_𝑖(𝐿+𝑈)𝑅_𝑗(𝐿+𝑈)+(1−𝛼)𝑆_𝑖(𝐿+𝑈)𝑆_𝑗(𝐿+𝑈),∀𝑖≠𝑗. (4.12) For 𝜔₁, we have 𝐴₂>2, it is in contradiction with ((4.8)b). Therefore, 𝜔₁ should be deleted.
Secondly, we prove (II).(1)When 0<𝜔≤1,  𝜎<0, 𝐴₂=2(1−𝜔)−2(𝜔−𝜎)𝜎𝛼𝑅_𝑖(𝐿)𝑅_𝑗(𝐿)+(1−𝛼)𝑆_𝑖(𝐿)𝑆_𝑗(𝐿)−𝜔𝜎𝛼𝑅_𝑖(𝐿)𝑅_𝑗(𝑈)+(1−𝛼)𝑆_𝑖(𝑈)𝑆_𝑗(𝐿)−𝜔𝜎𝛼𝑅_𝑖(𝑈)𝑅_𝑗(𝐿)+(1−𝛼)𝑆_𝑖(𝐿)𝑆_𝑗(𝑈,𝐴₃)=(1−𝜔)²−𝛼(𝜔−𝜎)𝑅_𝑖(𝐿)+𝜔𝑅_𝑖(𝑈)(𝜔−𝜎)𝑅_𝑗(𝐿)+𝜔𝑅_𝑗(𝑈−()𝑆)1−𝛼)(𝜔−𝜎_𝑖(𝐿)+𝜔𝑆_𝑖(𝑈)(𝜔−𝜎)𝑆_𝑗(𝐿)+𝜔𝑆_𝑗(𝑈). (4.13) By 𝐴₂−𝐴₃<𝐴₁, we have 4𝜎²𝛼𝑅_𝑖(𝐿)𝑅_𝑗(𝐿)+(1−𝛼)𝑆_𝑖(𝐿)𝑆_𝑗(𝐿)−2𝜔𝜎𝛼𝑅_𝑖(𝐿)𝑅_𝑗(𝐿+𝑈)+(1−𝛼)𝑆_𝑖(𝐿)𝑆_𝑗(𝐿+𝑈)+𝛼𝑅_𝑖(𝐿+𝑈)𝑅_𝑗(𝐿)+(1−𝛼)𝑆_𝑖(𝐿+𝑈)𝑆_𝑗(𝐿)+𝜔²𝛼𝑅_𝑖(𝐿+𝑈)𝑅_𝑗(𝐿+𝑈)+(1−𝛼)𝑆_𝑖(𝐿+𝑈)𝑆_𝑗(𝐿+𝑈)−1<0. (4.14) It is easy to verify that the discriminant Δ of the quadratic in 𝜎 satisfies Δ>0, and so there holds 𝜔𝑃^′₁−𝜔𝑃^′2₂+𝑃^′₃𝑃^′₃<𝜎<𝜔𝑃^′₁+𝜔𝑃^′2₂+𝑃^′₃𝑃^′₃. (4.15) By 𝜎<0 and 1−𝜎²[𝛼𝑅_𝑖(𝐿)𝑅_𝑗(𝐿)+(1−𝛼)𝑆_𝑖(𝐿)𝑆_𝑗(𝐿)]>0, forall𝑖,𝑗∈𝑁,𝑖≠𝑗, we obtain max_{𝑖,𝑗𝑖≠𝑗}𝜔𝑃^′₁−𝜔𝑃^′2₂+𝑃^′₃𝑃^′₃<𝜎<0. (4.16)(2)When 1<𝜔<𝑡, 𝜎<0, 𝐴₂=2(𝜔−1)−2(𝜔−𝜎)𝜎𝛼𝑅_𝑖(𝐿)𝑅_𝑗(𝐿)+(1−𝛼)𝑆_𝑖(𝐿)𝑆_𝑗(𝐿)−𝜔𝜎𝛼𝑅_𝑖(𝐿)𝑅_𝑗(𝑈)+(1−𝛼)𝑆_𝑖(𝑈)𝑆_𝑗(𝐿)−𝜔𝜎𝛼𝑅_𝑖(𝑈)𝑅_𝑗(𝐿)+(1−𝛼)𝑆_𝑖(𝐿)𝑆_𝑗(𝑈,𝐴₃)=(1−𝜔)²−𝛼(𝜔−𝜎)𝑅_𝑖(𝐿)+𝜔𝑅_𝑖(𝑈)(𝜔−𝜎)𝑅_𝑗(𝐿)+𝜔𝑅_𝑗(𝑈−()𝑆)1−𝛼)(𝜔−𝜎_𝑖(𝐿)+𝜔𝑆_𝑖(𝑈)(𝜔−𝜎)𝑆_𝑗(𝐿)+𝜔𝑆_𝑗(𝑈). (4.17) By 𝐴₂−𝐴₃<𝐴₁, we have 4𝜎²𝛼𝑅_𝑖(𝐿)𝑅_𝑗(𝐿)+(1−𝛼)𝑆_𝑖(𝐿)𝑆_𝑗(𝐿)−2𝜔𝜎𝛼𝑅_𝑖(𝐿)𝑅_𝑗(𝐿+𝑈)+(1−𝛼)𝑆_𝑖(𝐿)𝑆_𝑗(𝐿+𝑈)+𝛼𝑅_𝑖(𝐿+𝑈)𝑅_𝑗(𝐿)+(1−𝛼)𝑆_𝑖(𝐿+𝑈)𝑆_𝑗(𝐿)+𝜔²𝛼𝑅_𝑖(𝐿+𝑈)𝑅_𝑗(𝐿+𝑈)+(1−𝛼)𝑆_𝑖(𝐿+𝑈)𝑆_𝑗(𝐿+𝑈)−𝜔²+4𝜔−4<0. (4.18) It is easy to verify that the discriminant Δ of the quadratic in 𝜎 satisfies Δ>0, and so there holds 𝜔𝑃^′₁−𝜔²𝑃^′2₂+(𝜔−2)²𝑃^′₃𝑃^′₃<𝜎<𝜔𝑃^′₁+𝜔²𝑃^′2₂+(𝜔−2)²𝑃^′₃𝑃^′₃. (4.19) By 𝜎<0 and 1−𝜎²[𝛼𝑅_𝑖(𝐿)𝑅_𝑗(𝐿)+(1−𝛼)𝑆_𝑖(𝐿)𝑆_𝑗(𝐿)]>0, forall𝑖,𝑗∈𝑁,𝑖≠𝑗, we obtain max_{𝑖,𝑗𝑖≠𝑗}𝜔𝑃^′₁−𝜔²𝑃^′2₂+(𝜔−2)²𝑃^′₃𝑃^′₃<𝜎<0. (4.20) Therefore, by (4.16) and (4.20), we get max_{𝑖,𝑗𝑖≠𝑗}𝜔𝑃^′₁−𝜔²𝑃^′2₂+𝑃^′₃min𝜔²,(𝜔−2)²𝑃^′₃<𝜎<0. (4.21)
Finally, we prove (III).(1)When 0<𝜔≤1, 𝜎≥𝑡, 𝐴₂=2(1−𝜔)+2(𝜔−𝜎)𝜎𝛼𝑅_𝑖(𝐿)𝑅_𝑗(𝐿)+(1−𝛼)𝑆_𝑖(𝐿)𝑆_𝑗(𝐿)+𝜔𝜎𝛼𝑅_𝑖(𝐿)𝑅_𝑗(𝑈)+(1−𝛼)𝑆_𝑖(𝑈)𝑆_𝑗(𝐿)+𝜔𝜎𝛼𝑅_𝑖(𝑈)𝑅_𝑗(𝐿)+(1−𝛼)𝑆_𝑖(𝐿)𝑆_𝑗(𝑈,𝐴₃)=(1−𝜔)²−𝛼(𝜎−𝜔)𝑅_𝑖(𝐿)+𝜔𝑅_𝑖(𝑈)(𝜎−𝜔)𝑅_𝑗(𝐿)+𝜔𝑅_𝑗(𝑈−()𝑆)1−𝛼)(𝜎−𝜔_𝑖(𝐿)+𝜔𝑆_𝑖(𝑈)(𝜎−𝜔)𝑆_𝑗(𝐿)+𝜔𝑆_𝑗(𝑈). (4.22) By 𝐴₂−𝐴₃<𝐴₁, we have 4𝜎²𝛼𝑅_𝑖(𝐿)𝑅_𝑗(𝐿)+(1−𝛼)𝑆_𝑖(𝐿)𝑆_𝑗(𝐿)−2𝜔𝜎𝛼𝑅_𝑖(𝐿)𝑅_𝑗(𝐿)−𝑅_𝑗(𝑈)+(1−𝛼)𝑆_𝑖(𝐿)𝑆_𝑗(𝐿)−𝑆_𝑗(𝑈)−2𝜔𝜎𝛼𝑅_𝑗(𝐿)𝑅_𝑖(𝐿)−𝑅_𝑖(𝑈)+(1−𝛼)𝑆_𝑗(𝐿)𝑆_𝑖(𝐿)−𝑆_𝑖(𝑈)+𝜔²𝛼𝑅_𝑖(𝐿)−𝑅_𝑖(𝑈)𝑅_𝑗(𝐿)−𝑅_𝑗(𝑈)+(1−𝛼)𝑆_𝑖(𝐿)−𝑆_𝑖(𝑈)𝑆_𝑗(𝐿)−𝑆_𝑗(𝑈)−𝜔²<0. (4.23) It is easy to verify that the discriminant Δ of the quadratic in 𝜎 satisfies Δ>0, and so there holds 𝜔𝑃^′₄−𝜔𝑃^′2₅+𝑃^′₃𝑃^′₃<𝜎<𝜔𝑃^′₄+𝜔𝑃^′2₅+𝑃^′₃𝑃^′₃. (4.24) By 𝜎≥𝑡, we obtain 𝑡≤𝜎<min_{𝑖,𝑗𝑖≠𝑗}𝜔𝑃^′₄+𝜔𝑃^′2₅+𝑃^′₃𝑃^′₃. (4.25)(2)When 1<𝜔<𝑡,  𝜎≥𝑡, 𝐴₂=2(𝜔−1)+2(𝜎−𝜔)𝜎𝛼𝑅_𝑖(𝐿)𝑅_𝑗(𝐿)+(1−𝛼)𝑆_𝑖(𝐿)𝑆_𝑗(𝐿)+𝜔𝜎𝛼𝑅_𝑖(𝐿)𝑅_𝑗(𝑈)+(1−𝛼)𝑆_𝑖(𝑈)𝑆_𝑗(𝐿)+𝜔𝜎𝛼𝑅_𝑖(𝑈)𝑅_𝑗(𝐿)+(1−𝛼)𝑆_𝑖(𝐿)𝑆_𝑗(𝑈,𝐴₃)=(1−𝜔)²−𝛼(𝜎−𝜔)𝑅_𝑖(𝐿)+𝜔𝑅_𝑖(𝑈)(𝜎−𝜔)𝑅_𝑗(𝐿)+𝜔𝑅_𝑗(𝑈−()𝑆)1−𝛼)(𝜎−𝜔_𝑖(𝐿)+𝜔𝑆_𝑖(𝑈)(𝜎−𝜔)𝑆_𝑗(𝐿)+𝜔𝑆_𝑗(𝑈). (4.26) By 𝐴₂−𝐴₃<𝐴₁, we have 4𝜎²𝛼𝑅_𝑖(𝐿)𝑅_𝑗(𝐿)+(1−𝛼)𝑆_𝑖(𝐿)𝑆_𝑗(𝐿)−2𝜔𝜎𝛼𝑅_𝑖(𝐿)𝑅_𝑗(𝐿)−𝑅_𝑗(𝑈)+(1−𝛼)𝑆_𝑖(𝐿)𝑆_𝑗(𝐿)−𝑆_𝑗(𝑈)−2𝜔𝜎𝛼𝑅_𝑗(𝐿)𝑅_𝑖(𝐿)−𝑅_𝑖(𝑈)+(1−𝛼)𝑆_𝑗(𝐿)𝑆_𝑖(𝐿)−𝑆_𝑖(𝑈)+𝜔²𝛼𝑅_𝑖(𝐿)−𝑅_𝑖(𝑈)𝑅_𝑗(𝐿)−𝑅_𝑗(𝑈)+(1−𝛼)𝑆_𝑖(𝐿)−𝑆_𝑖(𝑈)𝑆_𝑗(𝐿)−𝑆_𝑗(𝑈)−𝜔²+4𝜔−4<0. (4.27) It is easy to verify that the discriminant Δ of the quadratic in 𝜎 satisfies Δ>0, and so there holds 𝜔𝑃₄−𝜔²𝑃^′2₅+(𝜔−2)²𝑃^′₃𝑃^′₃<𝜎<𝜔𝑃^′₄+𝜔²𝑃^′2₅+(𝜔−2)²𝑃^′₃𝑃^′₃. (4.28) By 𝜎≥𝑡, we obtain 𝑡≤𝜎<min_{𝑖,𝑗𝑖≠𝑗}𝜔𝑃^′₄+𝜔²𝑃^′2₅+(𝜔−2)²𝑃^′₃𝑃^′₃. (4.29) Therefore, by (4.25) and (4.29), we obtain 𝑡≤𝜎<min_{𝑖,𝑗𝑖≠𝑗}𝜔𝑃^′₄+𝜔²𝑃^′2₅+𝑃^′₃min𝜔²,(𝜔−2)²𝑃^′₃. (4.30)

We can obtain the following results easily.

Theorem 4.2. Let 𝐴∈𝐷𝐷(𝛼). If 𝑅_𝑖(𝐿+𝑈)𝑅_𝑗(𝐿+𝑈)−𝑆_𝑖(𝐿+𝑈)𝑆_𝑗(𝐿+𝑈)>0, when 0<𝜔≤1, the following conditions hold: (I)0≤𝛼<𝑅_𝑖(𝐿+𝑈)𝑅_𝑗(𝐿+𝑈)−𝑆_𝑖(𝐿+𝑈)𝑆_𝑗(𝐿+𝑈)𝑅_𝑖(𝐿+𝑈)𝑅_𝑗(𝐿+𝑈)−𝑆_𝑖(𝐿+𝑈)𝑆_𝑗(),()𝐿+𝑈II𝜔𝑃^′₁−𝜔²𝑃^′2₂+𝜔²𝑃^′₃𝑃^′₃<𝜔𝑃₁−𝜔²𝑃²₂+𝜔²𝑃₃𝑃₃,(III)𝜔𝑃^′₄+𝜔²𝑃^′2₅+𝜔²𝑃^′₃𝑃^′₃>𝜔𝑃₄+𝜔²𝑃²₅+𝜔²𝑃₃𝑃₃, (4.31) or when 1<𝜔<𝑡, the following conditions hold: (I)0≤𝛼<𝑅_𝑖(𝐿+𝑈)𝑅_𝑗(𝐿+𝑈)−𝑆_𝑖(𝐿+𝑈)𝑆_𝑗(𝐿+𝑈)𝑅_𝑖(𝐿+𝑈)𝑅_𝑗(𝐿+𝑈)−𝑆_𝑖(𝐿+𝑈)𝑆_𝑗(),()𝐿+𝑈II𝜔𝑃^′₁−𝜔²𝑃^′2₂+(𝜔−2)²𝑃^′₃𝑃^′₃<𝜔𝑃₁−𝜔²𝑃²₂+(𝜔−2)²𝑃₃𝑃₃,(III)𝜔𝑃^′₄+𝜔²𝑃^′2₅+(𝜔−2)²𝑃^′₃𝑃^′₃>𝜔𝑃₄+𝜔²𝑃²₅+(𝜔−2)²𝑃₃𝑃₃, (4.32) then the area of convergence of AOR method obtained by Theorem 4.1 is larger than that obtained by Theorem 2.5.

Theorem 4.3. Let 𝐴∈𝐷𝐷(𝛼). If 𝑅_𝑖(𝐿+𝑈)𝑅_𝑗(𝐿+𝑈)−𝑆_𝑖(𝐿+𝑈)𝑆_𝑗(𝐿+𝑈)<0, when 0<𝜔≤1, the following conditions hold: (I)𝑅_𝑖(𝐿+𝑈)𝑅_𝑗(𝐿+𝑈)−𝑆_𝑖(𝐿+𝑈)𝑆_𝑗(𝐿+𝑈)𝑅_𝑖(𝐿+𝑈)𝑅_𝑗(𝐿+𝑈)−𝑆_𝑖(𝐿+𝑈)𝑆_𝑗()()𝐿+𝑈<𝛼≤1,II𝜔𝑃^′₁−𝜔²𝑃^′2₂+𝜔²𝑃^′₃𝑃^′₃<𝜔𝑃₁−𝜔²𝑃²₂+𝜔²𝑃₃𝑃₃,(III)𝜔𝑃^′₄+𝜔²𝑃^′2₅+𝜔²𝑃^′₃𝑃^′₃>𝜔𝑃₄+𝜔²𝑃²₅+𝜔²𝑃₃𝑃₃, (4.33) or when 1<𝜔<𝑡, the following conditions hold: (I)𝑅_𝑖(𝐿+𝑈)𝑅_𝑗(𝐿+𝑈)−𝑆_𝑖(𝐿+𝑈)𝑆_𝑗(𝐿+𝑈)𝑅_𝑖(𝐿+𝑈)𝑅_𝑗(𝐿+𝑈)−𝑆_𝑖(𝐿+𝑈)𝑆_𝑗()()𝐿+𝑈<𝛼≤1,II𝜔𝑃^′₁−𝜔²𝑃^′2₂+(𝜔−2)²𝑃^′₃𝑃^′₃<𝜔𝑃₁−𝜔²𝑃²₂+(𝜔−2)²𝑃₃𝑃₃,(III)𝜔𝑃^′₄+𝜔²𝑃^′2₅+(𝜔−2)²𝑃^′₃𝑃^′₃>𝜔𝑃₄+𝜔²𝑃²₅+(𝜔−2)²𝑃₃𝑃₃, (4.34) then the area of convergence of AOR method obtained by Theorem 4.1 is larger than that obtained by Theorem 2.5.

5. Examples

In the following examples, we give the areas of convergence of AOR method to show that our results are better than ones obtained by Theorems 2.4 and 2.5.

Example 5.1 (see [6]). Let 𝐴=532263219=𝐷−𝑇−𝑆, (5.1) where 𝐷=,500060009,𝑇=000−200−2−10,𝑆=0−3−200−3000𝐿=𝐷⁻¹−1𝑇=0003−2009−190,𝑈=𝐷⁻¹3𝑆=0−5−25100−23000,𝐿+𝑈=0−5−25−1310−2−29−190. (5.2) Obviously, 𝐴∉𝑆𝐷, but 𝐴∈𝐷𝐷(1/2).

By Theorem 4.1, we have the following area of convergence: (1)0≤𝜎<1.1896,0<𝜔<max1.2390,2𝜎1+𝜌𝑀_𝜎,𝜎(2)√,or0<𝜔≤1,−1.0266𝜔<𝜎<0,or1<𝜔<1.2390,22𝜔−3201𝜔²−800𝜔+800(3)√20<𝜎<0,or0<𝜔≤1,1.2390<𝜎<2.0266𝜔,or1<𝜔<1.2390,1.2390≤𝜎<−2𝜔+3201𝜔²−800𝜔+800.20 (5.3) Obviously, 𝐴∈𝐷𝐷.

By Theorem 2.5, we have the following area of convergence: (1)0≤𝜎<1.1896,0<𝜔<max1.0455,2𝜎1+𝜌𝑀_𝜎,𝜎(2)√,or0<𝜔≤1,−0.6712𝜔<𝜎<0,or1<𝜔<1.0455,7𝜔−317𝜔²−64𝜔+648(3)√<𝜎<0,or0<𝜔≤1,1.0455≤𝜎<1.6712𝜔,or1<𝜔<1.0455,1.0455≤𝜎<𝜔+317𝜔²−64𝜔+648. (5.4) In addition, 𝐴∈𝐷(1/2).

By Theorem 2.4, we have the following area of convergence: (1)0≤𝜎<1.1896,0<𝜔<max1.1250,2𝜎1+𝜌𝑀_𝜎,𝜎(2)7,or0<𝜔≤1,−0.6𝜔<𝜎<0,or1<𝜔<1.1250,59𝜔−5(3)<𝜎<0,or0<𝜔≤1,1.1250<𝜎<1.31𝜔,or1<𝜔<1.1250,1.1250≤𝜎<1.8−0.49𝜔. (5.5)

Now we give two figures. In Figure 1, we can see that the area of convergence obtained by Theorem 4.1 (real line) is larger than that obtained by Theorem 2.5 (virtual line). In Figure 2, we can see that the area of convergence obtained by Theorem 4.1 (real line) is larger than that obtained by Theorem 2.4 (virtual line). From above we know that the area of convergence obtained by Theorem 4.1 is larger than ones obtained by Theorems 2.5 and 2.4.

Figure 1 

Figure 2 

Example 5.2. Let 𝐴=311121213. (5.6) Obviously, 𝐴∈𝐷𝐷(1/3), 𝐴∉𝐷(𝛼), 𝐴∉𝐷𝐷. So we cannot use Theorems 2.4 and 2.5. By Theorem 4.1, we have the following area of convergence: (1)0≤𝜎<1.0724,0<𝜔<max1.0693,2𝜎1+𝜌𝑀_𝜎,𝜎(2)√,or0<𝜔≤1,−0.1973𝜔<𝜎<0,or1<𝜔<1.0693,37𝜔−1945𝜔²−7776𝜔+7776(3)√36<𝜎<0,or0<𝜔≤1,1.0693<𝜎<1.1973𝜔,or1<𝜔<1.0693,1.0693≤𝜎<1945𝜔²−7776𝜔+7776−𝜔.36 (5.7)

Acknowledgments

The authors would like to thank the referees for their valuable comments and suggestions, which greatly improved the original version of this paper. This work was supported by the National Natural Science Foundation of China (Grant no. 11001144) and the Science and Technology Program of Shandong Universities of China (J10LA06).