# Improving Results on Convergence of AOR Method

Authors: Guangbin Wang; Ting Wang; Yanli Du
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

We present some sufficient conditions on convergence of AOR method for solvingAx=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.


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## Body

## 1. Introduction

Let us denote all complex square matrices byCn×n and all complex vectors by Cn.ForA=(aij)∈Cn×n, we denote by ρ(A) the spectral radius of matrix A.Let us consider linear systemAx=b, where b∈Cn is a given vector and x∈Cn 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
(1.1)L=D-1T,U=D-1S,
where det(D)≠0.Then the AOR method [1] can be written as
(1.2)xk+1=Mσ,ωxk+d,k=0,1,…,x0∈Cn,
where
(1.3)Mσ,ω=(I-σL)-1[(1-ω)I+(ω-σ)L+ωU],d=ω(I-σL)-1b,ω,σ∈R.


## 2. Preliminaries

We denote(2.1)Ri(A)=∑j≠i|aij|,Si(A)=∑j≠i|aji|,Pi,α(A)=αRi(A)+(1-α)Si(A),∀i∈N.
For any matrix A=(aij)∈Cn×n, the comparison matrix M(A)=(mij)∈Rn×n is defined by
(2.2)mii=|aii|,mij=-|aij|,i,j∈N,i≠j,N={1,2,…,n}.Definition 2.1 (see [2]).
A matrixA∈Cn×n is called a strictly diagonally dominant matrix (SD) if
(2.3)|aii|>Ri(A),∀i∈N.
A matrix A∈Cn×n is called a strictly doubly diagonally dominant matrix (DD) if
(2.4)|aii||ajj|>Ri(A)Rj(A),∀i,j∈N,i≠j.Definition 2.2 (see [3]).
A matrixA∈Cn×n is called a strictly α diagonally dominant matrix (D(α)) if there exits α∈[0,1], such that
(2.5)|aii|>αRi(A)+(1-α)Si(A),∀i∈N.Definition 2.3 (see [4]).
LetA=(aij)∈Cn×n, if there exits α∈[0,1] such that
(2.6)|aii||ajj|>αRi(A)Rj(A)+(1-α)Si(A)Sj(A),∀i,j∈N,i≠j,
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]).
LetA∈D(α), then AOR method converges for
(2.7)(I)0≤σ<2(1+ρ(M0,1(M(A))))=s,0<ω<max{2(1+maxiPi,α(L+U))=t,2σ(1+ρ(Mσ,σ))},or(II)maxi(-ω(1-Pi,α(L+U))+2max(0,ω-1))2Pi,α(L)<σ<0,0<ω<t,or(III)t≤σ<mini(ω(1+Pi,α(L)-Pi,α(U))+2min(0,1-ω))2Pi,α(L),0<ω<t.Theorem 2.5 (see [6]).
LetA∈DD, then AOR method converges for
(2.8)(I)0≤σ<2(1+ρ(M0,1(M(A))))=s,0<ω<max{2σ(1+ρ(Mσ,σ)),mini,ji≠j21+Ri(L+U)Rj(L+U)=t},or(II)maxi,ji≠jωP1-ω2P22+P3min(ω2,(ω-2)2)P3<σ<0,0<ω<t,or(III)t≤σ<mini,ji≠jωP4+ω2P52+P3min(ω2,(ω-2)2)P3,0<ω<t,
where
(2.9)P1=Ri(L)Rj(L+U)+Ri(L+U)Rj(L),P2=Ri(L)Rj(L+U)-Ri(L+U)Rj(L),P3=4Ri(L)Rj(L),P4=Ri(L)(Rj(L)-Rj(U))+Rj(L)(Ri(L)-Ri(U)),P5=Ri(L)(Rj(L)-Rj(U))-Rj(L)(Ri(L)-Ri(U)).


## 3. Upper Bound for Spectral Radius ofMσ,ω

In the following, we present an upper bound for spectral radius of AOR iterative matrixMσ,ω for strictly doubly α diagonally dominant coefficient matrix.Lemma 3.1 (see [4]).
IfA∈DD(α), then A is a nonsingular H-matrix.Theorem 3.2.
LetA∈DD(α), if 1-σ2[αRi(L)Rj(L)+(1-α)Si(L)Sj(L)]>0, for  all  j∈N,i≠j, then
(3.1)ρ(Mσ,ω)≤maxi,ji≠jA2+A22-4A1A32A1,
where
(3.2)A1=1-σ2[αRi(L)Rj(L)+(1-α)Si(L)Sj(L)],A2=2|1-ω|+2|ω-σ||σ|[αRi(L)Rj(L)+(1-α)Si(L)Sj(L)]+|ω||σ|[αRi(L)Rj(U)+(1-α)Si(U)Sj(L)]+|ω|σ||[αRi(U)Rj(L)+(1-α)Si(L)Sj(U)],A3=(1-ω)2-α[|ω-σ|Ri(L)+|ω|Ri(U)][|ω-σ|Rj(L)+|ω|Rj(U)]-(1-α)[|ω-σ|Si(L)+|ω|Si(U)][|ω-σ|Sj(L)+|ω|Sj(U)].Proof.
Letλ be an eigenvalue of Mσ,ω such that
(3.3)det(λI-Mσ,ω)=0,
that is,
(3.4)det(λ(I-σL)-((1-ω)I+(ω-σ)L+ωU))=0.
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 Mσ,ω, that is, if
(3.5)|λ-(1-ω)|2>α[∑t≠i|λσlit-(ω-σ)lit-ωuit|][∑l≠j|λσljl-(ω-σ)ljl-ωujl|]+(1-α)[∑k≠i|λσlki-(ω-σ)|lki-ωuki]×[∑s≠j|λσlsj-(ω-σ)|lsj-ωusj],∀i,j∈N,i≠j,
then λ is not an eigenvalue of Mσ,ω. Especially, if
(3.6)(|λ|-|1-ω|)2>α[∑t≠i(|λ||σ||lit|+|ω-σ||lit|+|ω||uit|)]×[∑l≠j(|λ||σ||ljl|+|ω-σ||ljl|+|ω||ujl|)]+(1-α)[∑k≠i(|λ||σ||lki|+|ω-σ||lki|+|ω|uki)]×[∑s≠j(|λ||σ||lsj|+|ω-σ||lsj|+|ω||usj|)],
then λ is not an eigenvalue of Mσ,ω.
Ifλ is an eigenvalue of Mσ,ω, then there exits at least a couple of i,j(i≠j), such that
(3.7)(|λ|-|1-ω|)2≤α[∑t≠i(|λ||σ||lit|+|ω-σ||lit|+|ω||uit|)]×[∑l≠j(|λ||σ||ljl|+|ω-σ||ljl|+|ω||ujl|)]+(1-α)[∑k≠i(|λ||σ||lki|+|ω-σ||lki|+|ω|uki)]×[∑s≠j(|λ||σ||lsj|+|ω-σ||lsj|+|ω||usj|)],
that is,
(3.8)A1|λ|2-A2|λ|+A3≤0.
Since A1=1-σ2[αRi(L)Rj(L)+(1-α)Si(L)Sj(L)]>0, and the discriminant Δ of the quadratic in |λ| satisfies Δ≥0, then the solution of (3.8) satisfies
(3.9)A2-A22-4A1A32A1≤|λ|≤A2+A22-4A1A32A1.
So
(3.10)ρ(Mσ,ω)≤maxi,ji≠jA2+A22-4A1A32A1.


## 4. Improving Results on Convergence of AOR Method

In this section, we present new results on convergence of AOR method.Theorem 4.1.
LetA∈DD(α), then AOR method converges if ω,σ satisfy either
(4.1)(I)0≤σ<2(1+ρ(M0,1(M(A))))=s,0<ω<max{21+αRi(L+U)Rj(L+U)+(1-α)Si(L+U)Sj(L+U)mini,ji≠j21+αRi(L+U)Rj(L+U)+(1-α)Si(L+U)Sj(L+U)2σ(1+ρ(Mσ,σ)),mini,ji≠j21+αRi(L+U)Rj(L+U)+(1-α)Si(L+U)Sj(L+U)=t},or(II)maxi,ji≠jωP1′-ω2P2'2+P3′min(ω2,(ω-2)2)P3′<σ<0,0<ω<t,or(III)t≤σ<mini,ji≠jωP4′+ω2P5'2+P3′min(ω2,(ω-2)2)P3′,0<ω<t,
where
(4.2)P1′=αRi(L)Rj(L+U)+(1-α)Si(L)Sj(L+U)+αRi(L+U)Rj(L)+(1-α)Si(L+U)Sj(L),P2′=αRi(L)Rj(L+U)+(1-α)Si(L)Sj(L+U)-αRi(L+U)Rj(L)-(1-α)Si(L+U)Sj(L),P3′=4[αRi(L)Rj(L)+(1-α)Si(L)Sj(L)],P4′=αRi(L)[Rj(L)-Rj(U)]+(1-α)Si(L)[Sj(L)-Sj(U)]+αRj(L)[Ri(L)-Ri(U)]+(1-α)Sj(L)[Si(L)-Si(U)],P5′=αRi(L)[Rj(L)-Rj(U)]+(1-α)Si(L)[Sj(L)-Sj(U)]-αRj(L)[Ri(L)-Ri(U)]-(1-α)Sj(L)[Si(L)-Si(U)].Proof.
It is easy to verify that for eachσ, which satisfies one of the conditions (I)–(III), we have
(4.3)1-σ2[αRi(L)Rj(L)+(1-α)Si(L)Sj(L)]>0,∀i≠j;i,j∈N.
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(Mσ,σ)<1 holds for 0≤σ<s and for σ≠0,
(4.4)Mσ,ω=(1-ωσ)I+(ωσ)Mσ,σ.
If 0<ω<max{2σ/(1+ρ(Mσ,σ)),t}=2σ/(1+ρ(Mσ,σ)), then
(4.5)0<ωσ<2(1+ρ(Mσ,σ)),
by extrapolation theorem [6], we have ρ(Mσ,ω)<1.
If0<ω<max{2σ/(1+ρ(Mσ,σ)),t}=t, then it remains to analyze the case
(4.6)2σ(1+ρ(Mσ,σ))≤ω<t,0≤σ<s.
Since when ρ(Mσ,σ)<1,
(4.7)σ<2σ(1+ρ(Mσ,σ)),
then 0≤σ<ω. From
(4.8)A2-A3<A1(a)A2<2A1≤2(b)⟺A2+A22-4A1A32A1<1,A22-4A1A3<2A1≤2(c)
we have ρ(Mσ,ω)≤maxi,ji≠j((A2+A22-4A1A3)/2A1)<1.(1)
Whenω≤1, it easy to verify that (4.8) holds.(2)
Whenω>1, since
(4.9)A1=1-σ2[αRi(L)Rj(L)+(1-α)Si(L)Sj(L)],A2=2(ω-σ)[αRi(L)Rj(L)+(1-α)Si(L)Sj(L)]+ωσ[αRi(L)Rj(U)+(1-α)Si(U)Sj(L)]+ωσ[αRi(U)Rj(L)+(1-α)Si(L)Sj(U)]+2|1-ω|,A3=(1-ω)2-α[(ω-σ)Ri(L)+ωRi(U)][(ω-σ)Rj(L)+ωRj(U)]-(1-α)[(ω-σ)Si(L)+ωSi(U)][(ω-σ)Sj(L)+ωSj(U)],then by A2-A3<A1 and 1-σ2[αRi(L)Rj(L)+(1-α)Si(L)Sj(L)]>0, foralli,j∈N,i≠j, we have
(4.10)ω2[1-αRi(L+U)Rj(L+U)-(1-α)Si(L+U)Sj(L+U)]-4ω+4>0.
It is easy to verify that the discriminant Δ of the quadratic in ω satisfies Δ>0, and so there holds
(4.11)ω1>21-αRi(L+U)Rj(L+U)-(1-α)Si(L+U)Sj(L+U),
or
(4.12)ω2<21+αRi(L+U)Rj(L+U)+(1-α)Si(L+U)Sj(L+U),∀i≠j.
For ω1, we have A2>2, it is in contradiction with ((4.8)b). Therefore, ω1 should be deleted.
Secondly, we prove(II).(1)
When0<ω≤1,  σ<0,
(4.13)A2=2(1-ω)-2(ω-σ)σ[αRi(L)Rj(L)+(1-α)Si(L)Sj(L)]-ωσ[αRi(L)Rj(U)+(1-α)Si(U)Sj(L)]-ωσ[αRi(U)Rj(L)+(1-α)Si(L)Sj(U)],A3=(1-ω)2-α[(ω-σ)Ri(L)+ωRi(U)][(ω-σ)Rj(L)+ωRj(U)]-(1-α)[(ω-σ)Si(L)+ωSi(U)][(ω-σ)Sj(L)+ωSj(U)].
By A2-A3<A1, we have
(4.14)4σ2[αRi(L)Rj(L)+(1-α)Si(L)Sj(L)]-2ωσ[αRi(L)Rj(L+U)+(1-α)Si(L)Sj(L+U)+αRi(L+U)Rj(L)+(1-α)Si(L+U)Sj(L)]+ω2[αRi(L+U)Rj(L+U)+(1-α)Si(L+U)Sj(L+U)-1]<0.
It is easy to verify that the discriminant Δ of the quadratic in σ satisfies Δ>0, and so there holds
(4.15)ωP1′-ωP2'2+P3′P3′<σ<ωP1′+ωP2'2+P3′P3′.
By σ<0 and 1-σ2[αRi(L)Rj(L)+(1-α)Si(L)Sj(L)]>0, foralli,j∈N,i≠j, we obtain
(4.16)maxi,ji≠jωP1′-ωP2'2+P3′P3′<σ<0.(2)
When1<ω<t, σ<0,
(4.17)A2=2(ω-1)-2(ω-σ)σ[αRi(L)Rj(L)+(1-α)Si(L)Sj(L)]-ωσ[αRi(L)Rj(U)+(1-α)Si(U)Sj(L)]-ωσ[αRi(U)Rj(L)+(1-α)Si(L)Sj(U)],A3=(1-ω)2-α[(ω-σ)Ri(L)+ωRi(U)][(ω-σ)Rj(L)+ωRj(U)]-(1-α)[(ω-σ)Si(L)+ωSi(U)][(ω-σ)Sj(L)+ωSj(U)].
By A2-A3<A1, we have
(4.18)4σ2[αRi(L)Rj(L)+(1-α)Si(L)Sj(L)]-2ωσ[αRi(L)Rj(L+U)+(1-α)Si(L)Sj(L+U)+αRi(L+U)Rj(L)+(1-α)Si(L+U)Sj(L)]+ω2[αRi(L+U)Rj(L+U)+(1-α)Si(L+U)Sj(L+U)]-ω2+4ω-4<0.
It is easy to verify that the discriminant Δ of the quadratic in σ satisfies Δ>0, and so there holds
(4.19)ωP1′-ω2P2'2+(ω-2)2P3′P3′<σ<ωP1′+ω2P2'2+(ω-2)2P3′P3′.
By σ<0 and 1-σ2[αRi(L)Rj(L)+(1-α)Si(L)Sj(L)]>0, foralli,j∈N,i≠j, we obtain
(4.20)maxi,ji≠jωP1′-ω2P2'2+(ω-2)2P3′P3′<σ<0.
Therefore, by (4.16) and (4.20), we get
(4.21)maxi,ji≠jωP1′-ω2P2'2+P3′min(ω2,(ω-2)2)P3′<σ<0.
Finally, we prove(III).(1)
When0<ω≤1, σ≥t,
(4.22)A2=2(1-ω)+2(ω-σ)σ[αRi(L)Rj(L)+(1-α)Si(L)Sj(L)]+ωσ[αRi(L)Rj(U)+(1-α)Si(U)Sj(L)]+ωσ[αRi(U)Rj(L)+(1-α)Si(L)Sj(U)],A3=(1-ω)2-α[(σ-ω)Ri(L)+ωRi(U)][(σ-ω)Rj(L)+ωRj(U)]-(1-α)[(σ-ω)Si(L)+ωSi(U)][(σ-ω)Sj(L)+ωSj(U)].
By A2-A3<A1, we have
(4.23)4σ2[αRi(L)Rj(L)+(1-α)Si(L)Sj(L)]-2ωσ[αRi(L)(Rj(L)-Rj(U))+(1-α)Si(L)(Sj(L)-Sj(U))]-2ωσ[αRj(L)(Ri(L)-Ri(U))+(1-α)Sj(L)(Si(L)-Si(U))]+ω2[α(Ri(L)-Ri(U))(Rj(L)-Rj(U))+(1-α)(Si(L)-Si(U))(Sj(L)-Sj(U))]-ω2<0.
It is easy to verify that the discriminant Δ of the quadratic in σ satisfies Δ>0, and so there holds
(4.24)ωP4′-ωP5'2+P3′P3′<σ<ωP4′+ωP5'2+P3′P3′.
By σ≥t, we obtain
(4.25)t≤σ<mini,ji≠jωP4′+ωP5'2+P3′P3′.(2)
When1<ω<t,  σ≥t,
(4.26)A2=2(ω-1)+2(σ-ω)σ[αRi(L)Rj(L)+(1-α)Si(L)Sj(L)]+ωσ[αRi(L)Rj(U)+(1-α)Si(U)Sj(L)]+ωσ[αRi(U)Rj(L)+(1-α)Si(L)Sj(U)],A3=(1-ω)2-α[(σ-ω)Ri(L)+ωRi(U)][(σ-ω)Rj(L)+ωRj(U)]-(1-α)[(σ-ω)Si(L)+ωSi(U)][(σ-ω)Sj(L)+ωSj(U)].
By A2-A3<A1, we have
(4.27)4σ2[αRi(L)Rj(L)+(1-α)Si(L)Sj(L)]-2ωσ[αRi(L)(Rj(L)-Rj(U))+(1-α)Si(L)(Sj(L)-Sj(U))]-2ωσ[αRj(L)(Ri(L)-Ri(U))+(1-α)Sj(L)(Si(L)-Si(U))]+ω2[α(Ri(L)-Ri(U))(Rj(L)-Rj(U))+(1-α)(Si(L)-Si(U))(Sj(L)-Sj(U))]-ω2+4ω-4<0.
It is easy to verify that the discriminant Δ of the quadratic in σ satisfies Δ>0, and so there holds
(4.28)ωP4-ω2P5'2+(ω-2)2P3′P3′<σ<ωP4′+ω2P5'2+(ω-2)2P3′P3′.
By σ≥t, we obtain
(4.29)t≤σ<mini,ji≠jωP4′+ω2P5'2+(ω-2)2P3′P3′.
Therefore, by (4.25) and (4.29), we obtain
(4.30)t≤σ<mini,ji≠jωP4′+ω2P5'2+P3′min(ω2,(ω-2)2)P3′.We can obtain the following results easily.Theorem 4.2.
LetA∈DD(α). If Ri(L+U)Rj(L+U)-Si(L+U)Sj(L+U)>0, when 0<ω≤1, the following conditions hold:
(4.31)(I)0≤α<Ri(L+U)Rj(L+U)-Si(L+U)Sj(L+U)Ri(L+U)Rj(L+U)-Si(L+U)Sj(L+U),(II)ωP1′-ω2P2'2+ω2P3′P3′<ωP1-ω2P22+ω2P3P3,(III)ωP4′+ω2P5'2+ω2P3′P3′>ωP4+ω2P52+ω2P3P3,
or when 1<ω<t, the following conditions hold:
(4.32)(I)0≤α<Ri(L+U)Rj(L+U)-Si(L+U)Sj(L+U)Ri(L+U)Rj(L+U)-Si(L+U)Sj(L+U),(II)ωP1′-ω2P2'2+(ω-2)2P3′P3′<ωP1-ω2P22+(ω-2)2P3P3,(III)ωP4′+ω2P5'2+(ω-2)2P3′P3′>ωP4+ω2P52+(ω-2)2P3P3,
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.
LetA∈DD(α). If Ri(L+U)Rj(L+U)-Si(L+U)Sj(L+U)<0, when 0<ω≤1, the following conditions hold:
(4.33)(I)Ri(L+U)Rj(L+U)-Si(L+U)Sj(L+U)Ri(L+U)Rj(L+U)-Si(L+U)Sj(L+U)<α≤1,(II)ωP1′-ω2P2'2+ω2P3′P3′<ωP1-ω2P22+ω2P3P3,(III)ωP4′+ω2P5'2+ω2P3′P3′>ωP4+ω2P52+ω2P3P3,
or when 1<ω<t, the following conditions hold:
(4.34)(I)Ri(L+U)Rj(L+U)-Si(L+U)Sj(L+U)Ri(L+U)Rj(L+U)-Si(L+U)Sj(L+U)<α≤1,(II)ωP1′-ω2P2'2+(ω-2)2P3′P3′<ωP1-ω2P22+(ω-2)2P3P3,(III)ωP4′+ω2P5'2+(ω-2)2P3′P3′>ωP4+ω2P52+(ω-2)2P3P3,
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 Theorems2.4 and 2.5.Example 5.1 (see [6]).
Let(5.1)A=(532263219)=D-T-S,
where
(5.2)D=(500060009),T=(000-200-2-10),S=(0-3-200-3000),L=D-1T=(000-1300-29-190),U=D-1S=(0-35-2500-12000),L+U=(0-35-25-130-12-29-190).
Obviously, A∉SD, but A∈DD(1/2).By Theorem4.1, we have the following area of convergence:
(5.3)(1)0≤σ<1.1896,0<ω<max{1.2390,2σ(1+ρ(Mσ,σ))},or(2)0<ω≤1,-1.0266ω<σ<0,or1<ω<1.2390,(22ω-3201ω2-800ω+800)20<σ<0,or(3)0<ω≤1,1.2390<σ<2.0266ω,or1<ω<1.2390,1.2390≤σ<(-2ω+3201ω2-800ω+800)20.
Obviously, A∈DD.By Theorem2.5, we have the following area of convergence:
(5.4)(1)0≤σ<1.1896,0<ω<max{1.0455,2σ(1+ρ(Mσ,σ))},or(2)0<ω≤1,-0.6712ω<σ<0,or1<ω<1.0455,(7ω-317ω2-64ω+64)8<σ<0,or(3)0<ω≤1,1.0455≤σ<1.6712ω,or1<ω<1.0455,1.0455≤σ<(ω+317ω2-64ω+64)8.
In addition, A∈D(1/2).By Theorem2.4, we have the following area of convergence:
(5.5)(1)0≤σ<1.1896,0<ω<max{1.1250,2σ(1+ρ(Mσ,σ))},or(2)0<ω≤1,-0.6ω<σ<0,or1<ω<1.1250,75ω-95<σ<0,or(3)0<ω≤1,1.1250<σ<1.31ω,or1<ω<1.1250,1.1250≤σ<1.8-0.49ω.Now we give two figures. In Figure1, 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 1Figure 2Example 5.2.
Let(5.6)A=(311121213).
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:
(5.7)(1)0≤σ<1.0724,0<ω<max{1.0693,2σ(1+ρ(Mσ,σ))},or(2)0<ω≤1,-0.1973ω<σ<0,or1<ω<1.0693,(37ω-1945ω2-7776ω+7776)36<σ<0,or(3)0<ω≤1,1.0693<σ<1.1973ω,or1<ω<1.0693,1.0693≤σ<(1945ω2-7776ω+7776-ω)36.



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*Source: 274636-2012-10-30.xml*