MPEMathematical Problems in Engineering1563-51471024-123XHindawi Publishing Corporation34198210.1155/2010/341982341982Research ArticleConvergence Analysis of Preconditioned AOR Iterative Method for Linear SystemsLiu Qingbing1, 2ChenGuoliang2GonçalvesPaulo Batista1Department of MathematicsZhejiang Wanli UniversityNingbo 315100Chinazwu.edu.cn2Department of MathematicsEast China Normal UniversityShanghai 200241Chinaecnu.edu.cn20101006201020102606200921022010130520102010Copyright © 2010This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

M-(H-)matrices appear in many areas of science and engineering, for example, in the solution of the linear complementarity problem (LCP) in optimization theory and in the solution of large systems for real-time changes of data in fluid analysis in car industry. Classical (stationary) iterative methods used for the solution of linear systems have been shown to convergence for this class of matrices. In this paper, we present some comparison theorems on the preconditioned AOR iterative method for solving the linear system. Comparison results show that the rate of convergence of the preconditioned iterative method is faster than the rate of convergence of the classical iterative method. Meanwhile, we apply the preconditioner to H-matrices and obtain the convergence result. Numerical examples are given to illustrate our results.

1. Introduction

In numerical linear algebra, the theory of M- and H-matrices is very important for the solution of linear systems of algebra equations by iterative methods (see, e.g., ). For example, (a) in the linear complementarity problem (LCP) (see [5, Section 10.1] for specific applications), where we are interested in finding a zRn such that z0, Mz+q0, zT(Mz+q)=0, with MRn×n and qRn given, a sufficient condition for a solution to exist, and to be found by a modification of an iterative method, especially of SOR, is that M is an H-matrix, with mi,i>0, i=1,,n ; (b) in fluid analysis, in the car modeling design [16, 17], it was observed that large linear systems with an H-matrix coefficient A are solved iteratively much faster if A is postmultiplied by a suitable diagonal matrix D, with di,i>0, i=1,,n, so that AD is strictly diagonally dominant. We consider the following linear system:Ax=b, where A is an n×n square matrix, x and b are two n-dimensional vectors. For any splitting, A=M-N with the nonsingular matrix M, the basic iterative method for solving the linear system (1.1) is as follows:xi+1=M-1Nxi+M-1b,i=0,1,2,.

Without loss of generality, let A=I-L-U and ai,10, i=2,,n, where L and U are strictly lower triangular and strictly upper triangular matrices of A, respectively. Then the iterative matrix of the AOR iterative method  for solving the linear system (1.1) isTγ,ω=(I-γL)-1[(1-ω)I+(ω-γ)L+ωU], where ω and γ are nonnegative real parameters with ω0.

To improve the convergence rate of the basic iterative methods, several preconditioned iterative methods have been proposed in [8, 12, 13, 1924]. We now transform the original system (1.1) into the preconditioned formPAx=Pb, where P is a nonsingular matrix. The corresponding basic iterative method is given in general by xi+1=MP-1NPxi+MP-1Pb,i=0,1,2,, where PA=MP-NP is a splitting of PA.

Milaszewicz  presented a modified Jacobi and Gauss-Seidel iterative methods by using the preconditioned matrix P=I+S, where P=(I+S)=[100-a2110-an101].

The author  suggests that if the original iteration matrix is nonnegative and irreducible, then performing Gaussian elimination on a selected column of iteration matrix to make it zero will improve the convergence of the iteration matrix.

In 2003, Hadjidimos et al.  considered the generalized preconditioner used in this case is of the formP(α)=(I+Sα)=[100-α2a2110-αnan101], where α=(α2,,αn)TRn-1 with αi[0,1], i=2,,n, constants. The selection of α's will be made from the (n-1)-dimensional nonnegative cone Kn-1 in such a way that none of the diagonal elements of the preconditioned matrix Ã=P(α)A vanishes. They discussed the convergence of preconditioned Jacobi and Gauss-Seidel when a coefficient matrix A is an M-matrix.

In this paper, we consider the preconditioned linear system of the formÃx=b̃, where Ã=(I+Sα)A and b̃=(I+Sα)b. It is clear that SαL=0. Thus, we obtain the equalityÃ=(I+Sα)A=(I+Sα)(I-L-U)=I-SD-L-SL+Sα-U-SU, where SD,SL, and SU are the diagonal, strictly lower, and strictly upper triangular parts of the matrix SαU, respectively. If we apply the AOR iterative method to the preconditioned linear system (1.8), then we get the preconditioned AOR iterative method whose iteration matrix isT̃γ,ω=(D̃-γL̃)-1[(1-ω)D̃+(ω-γ)L̃+ωŨ].

This paper is organized as follows. Section 2 is preliminaries. Section 3 will discuss the convergence of the preconditioned AOR method and obtain comparison theorems with the classical iterative method when a coefficient matrix is a Z-matrix. In Section 4 we apply the preconditioner to H-matrices and obtain the convergence result. In Section 5 we use numerical examples to illustrate our results.

2. Preliminaries

We say that a vector x is nonnegative (positive), denoted x0(x>0), if all its entries are nonnegative (positive). Similarly, a matrix B is said to be nonnegative, denoted B0, if all its entries are nonnegative or, equivalently, if it leaves invariant the set of all nonnegative vectors. We compare two matrices AB, when A-B0, and two vectors xy(x>y) when x-y0(x-y>0). Given a matrix A=(ai,j), we define the matrix |A|=(|ai,j|). It follows that |A|0 and that |AB||A||B| for any two matrices A and B of compatible size.

Definition 2.1.

A matrix A=(ai,j)Rn×n is called a Z-matrix if ai,j0 for ij. A matrix A is called a nonsingular M-matrix if A is a Z-matrix and A-10.

Definition 2.2.

A matrix A is an H-matrix if its comparison matrix A=(a¯i,j) is an M-matrix, where a¯i,j is a¯i,i=|ai,i|,a¯i,j=-|ai,j|,ij.

Definition 2.3 (see [<xref ref-type="bibr" rid="B3">1</xref>]).

The splitting A=M-N is called an H-splitting if M-|N| is an M-matrix and an H-compatible splitting if A=M-|N|.

Definition 2.4.

Let A=(ai,j)Rn×n. A=M-N is called a splitting of A if M is a nonsingular matrix. The splitting is called

convergent if ρ(M-1N)<1

regular if M-10 and N0

nonnegative if M-1N0

M-splitting if M is a nonsingular M-matrix and N0.

Lemma 2.5 (see [<xref ref-type="bibr" rid="B3">1</xref>]).

Let A=M-N be a splitting. If the splitting is an H-splitting, then A and M are H-matrices and ρ(M-1N)ρ(M-1|N|)<1. If the splitting is an H-compatible splitting and A is an H-matrix, then it is an H-splitting and thus convergent.

Lemma 2.6 (Perron-Frobenius theorem).

Let A0 be an irreducible matrix. Then the following hold:

A has a positive eigenvalue equal to ρ(A).

A has an eigenvector x>0 corresponding to ρ(A).

ρ(A) is a simple eigenvalue of A.

Lemma 2.7 (see [<xref ref-type="bibr" rid="B8">3</xref>, <xref ref-type="bibr" rid="B7">25</xref>]).

Let A=M-N be an M-splitting of A. Then ρ(M-1N)<1(=1) if and only if A is a nonsingular (singular) M-matrix. If A is irreducible, then here is a positive vector x such that M-1Nx=ρ(M-1N)x.

Lemma 2.8 (see [<xref ref-type="bibr" rid="B10">5</xref>]).

Let A0 be a nonnegative matrix. Then the following hold.

If Axβx for a vector x0 and x0, then ρ(A)β.

If Axγx for a vector x>0, then ρ(A)γ; moreover, if A is irreducible and if βxAxγx, equality excluded, for a vector x0 and x0, then β<ρ(A)<γ and x>0.

3. Convergence Theorems for <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M163"><mml:mrow><mml:mi>Z</mml:mi></mml:mrow></mml:math></inline-formula>-Matrix

We first consider the convergence of the iteration matrix T̃γ,ω of the preconditioned linear system (1.8) when the coefficient matrix is a Z-matrix.

Particularly, we consider αi=1, i=2,,n. DefineA¯=(I+S1)A=(I+S1)(I-L-U)=I-D-L-L+S1-U-U, where D,L, and U are diagonal, strictly lower triangular, and strictly upper triangular parts of the matrix S1U, respectively. Then the preconditioned AOR method is expressed as follows: T¯γ,ω=(D¯-γL¯)-1[(1-ω)D¯+(ω-γ)L¯+ωU¯], where D¯=I-D, L¯=L+L-S1, and U¯=U+U are the diagonal, strictly lower, and strictly upper triangular matrices obtained from A¯, respectively.

Lemma 3.1.

Let A=(ai,j)Rn×n be a Z-matrix. Then (I+Sα)A is also a Z-matrix.

Proof.

Since Ã=(ãi,j)=(I+Sα)A, we have ãi,j={ai,j,i=1,ai,j-αiai,1a1,j,i=2,,n. It is clear that Ã is a Z-matrix for any αi[0,1], i=2,,n.

Lemma 3.2.

Let Tγ,ω and T̃γ,ω be defined by (1.3) and (1.10). Assume that 0γω1(ω0,γ1). If A is an irreducible Z-matrix with ai1a1i<1, i=2,,n, for αi(0,1), i=2,,n, then Tγ,ω and T̃γ,ω are nonnegative and irreducible.

Proof.

Since A=I-L-U is irreducible. Then for αi(0,1), i=2,,n, we have that Ã=(I+Sα)A=D̃-L̃-Ũ is also irreducible. Observe that Tγ,ω=(1-ω)I+ω(1-γ)L+ωU+T, where T is a nonnegative matrix. As A is an irreducible Z-matrix and ω0, γ1, it is easy to show that Tγ,ω is nonnegative and irreducible. By assumption, D̃,L̃,and  Ũ are all nonnegative and thus T̃γ,ω is nonnegative. Observe that T̃γ,ω can be expressed as T̃γ,ω=(1-ω)I+ω(1-γ)D̃-1L̃+ωD̃-1Ũ+T̃, where T̃ is a nonnegative matrix. Since ω0, γ1, and Ã is irreducible, ω(1-γ)D̃-1L̃+ωD̃-1Ũ is irreducible. Hence, T̃γ,ω is irreducible from (3.5).

Our main result in this section is as follows.

Theorem 3.3.

Let Tγ,ω and T̃γ,ω be defined by (1.3) and (1.10). Assume that 0γω1(ω0,γ1). If A is an irreducible Z-matrix with ai1a1i<1, i=2,,n, for αi(0,1), i=2,,n, then

for αi(0,1), ρ(T̃γ,ω)<ρ(Tγ,ω)<1 if ρ(Tγ,ω)<1;

for αi[0,1], ρ(T̃γ,ω)=ρ(Tγ,ω)=1 if ρ(Tγ,ω)=1;

for αi(0,1), ρ(T̃γ,ω)>ρ(Tγ,ω)>1 if ρ(Tγ,ω)>1.

Proof.

Let A=I-L-U be irreducible. It is clear that I-γL is an M-matrix and (1-ω)I+(ω-γ)L+ωU0. So A=(I-γL)-[(1-ω)I+(ω-γ)L+ωU] is an M-splitting of A. From Lemma 2.7, there exists a positive vector x such that Tγ,ωx=λx, where λ denotes the spectral radius of Tγ,ω. Observe that Tγ,ω=(I-γL)-1[(1-ω)I+(ω-γ)L+ωU]; we have [(1-ω)I+(ω-γ)L+ωU]x=λ(I-γL)x, which is equivalent to (λ-1)(I-γL)x=ω(L+U-I)x. Let SαU=SD+SL+SU, where SD,SL, and SU are the diagonal, strictly lower, and strictly upper triangular parts of SαU, respectively. It is clear that SαL=0, so Ã=D̃-L̃-Ũ=(I-SD)-(L+SL-Sα)-(U+SU), where D̃=I-SD,L̃=L+SL-Sα,Ũ=U+SU. From (3.8) and (3.10), we have T̃γ,ωx-λx=(D̃-γL̃)-1[(1-ω)D̃+(ω-γ)L̃+ωŨ-λ(D̃-γL̃)]x=(D̃-γL̃)-1[(1-ω-λ)D̃+(ω-γ+λγ)L̃+ωŨ]x=(D̃-γL̃)-1[(1-ω-λ)(I-SD)+(ω-γ+λγ)(L+SL-Sα)+ω(U+SU)=(D̃-γL̃)-1[(1-ω-λ)I+(ω-γ+λγ)L+ωU-(1-ω-λ)SD+(ω-γ+λγ)SL-(ω-γ+λγ)Sα+ωSU]x=(D̃-γL̃)-1[(λ-1)SD+ωSD+(λ-1)γSL+ωSL+ωSU-(ω-γ+λγ)Sα]x=(D̃-γL̃)-1[(λ-1)SD+(λ-1)γSL+ωSαU-ωSα+ωSαL-(λ-1)γSα]x=(D̃-γL̃)-1[(λ-1)SD+(λ-1)γSL-(λ-1)γSα+ωSα(U+L-I)]x=(D̃-γL̃)-1[(λ-1)SD+(λ-1)γSL-(λ-1)γSα+(λ-1)Sα(I-γL)]x=(D̃-γL̃)-1[(λ-1)SD+(λ-1)γSL-(λ-1)γSα+(λ-1)Sα]x=(λ-1)(D̃-γL̃)-1[SD+(1-γ)Sα+γSL]x. Since ai,1a1,i<1, i=2,,n, then D̃-γL̃ is an M-matrix. Notice that SD0,  Sα0, and SL0. If λ<1, then from (3.11), we have T̃γ,ωxλx. As x>0, Lemma 2.8 implied that ρ(T̃γ,ω)λ=ρ(Tγ,ω). For the case of λ=1 and λ>1, T̃γ,ωx=λx and T̃γ,ωxλx are obtained from (3.11), respectively. Hence, Theorem 3.3 follows from Lemmas 2.8 and 3.2.

We next consider the case of αi=1, i=2,,n; the convergence theorem is given as follows see [26, 27].

Theorem 3.4.

Let Tγ,ω and T¯γ,ω be defined by (1.3) and (1.10). Assume that A is an irreducible Z-matrix and A(2:n,2:n) is an irreducible submatrix of A deleting the first row and the first column. Then for 0γω1(ω0,γ1) and ai1a1i<1, i=2,,n, we have

ρ(T¯γ,ω)<ρ(Tγ,ω)<1 if ρ(Tγ,ω)<1;

ρ(T¯γ,ω)=ρ(Tγ,ω)=1 if ρ(Tγ,ω)=1;

ρ(T¯γ,ω)>ρ(Tγ,ω)>1 if ρ(Tγ,ω)>1.

Proof.

Let A=I-L-U be irreducible. It is clear that I-γL is an M-matrix and (1-ω)I+(ω-γ)L+ωU0. So A=(I-γL)-[(1-ω)I+(ω-γ)L+ωU] is an M-splitting of A. From Lemma 2.7, there exists a positive vector x such that Tγ,ωx=λx, where λ denotes the spectral radius of Tγ,ω. Observe that Tγ,ω=(I-γL)-1[(1-ω)I+(ω-γ)L+ωU]; we have [(1-ω)I+(ω-γ)L+ωU]x=λ(I-γL)x, which is equivalent to (λ-1)(I-γL)x=ω(L+U-I)x. Similar to the proof of the equality (3.11), we have T¯γ,ωx-λx=(D¯-γL¯)-1[(1-ω)D¯+(ω-γ)L¯+ωU¯-λ(D¯-γL¯)]x=(D¯-γL¯)-1[(1-ω-λ)D¯+(ω-γ+λγ)L¯+ωU¯]x. Since D¯=I-D, L¯=L+L-S1, and U¯=U+U, then we have T¯γ,ωx-λx=(λ-1)(D¯-γL¯)-1(D+γL+(1-γ)S1)x. By computation, we have T¯γ,ω=(1-ω)I+ω(1-γ)D¯-1L¯+ωD¯-1U¯+H¯=[1-ωT¯1,20T¯2,2], where H¯ is a nonnegative matrix, T¯1,20 is a 1×(n-1) matrix, and T¯2,20 is an (n-1)×(n-1) matrix. As A is irreducible, then at least one a1,i0 and T¯1,2 is nonzero. Since A(2:n,2:n) is irreducible, it is clear that A¯(2:n,2:n) is irreducible. Since ω0 and γ1, from (3.17), we have that T¯2,2 is irreducible. Let u=(D¯-γL¯)-1(D+γL+(1-γ)S1)x,v=(D¯-γL¯)-1u. From (3.18), and x>0, we know that u0, and the first component of u is zero. Hence v0 and its first component is zero. Let x=(x1x2),v=(0v2), where x1R1>0, x2Rn-1>0, and v2Rn-10 being a nonzero vector. From (3.16) and (3.17), we have T¯γ,ωx-λx=(λ-1)v. That is, (1-ω)x1+T¯1,2x2=λx1,T¯2,2x2-λx2=(λ-1)v2.   If λ<1, from (3.22) and v2 is a nonzero vector, we have T¯2,2x2<λx2,(λ-1)v20. Since T¯2,2 is irreducible, from Lemma 2.8, we have ρ(T¯2,2)<λ. Since x2>0 and T¯1,2 is a nonzero nonnegative vector, from (3.21), we have (1-ω)x1<λx1. Namely, 1-ω<λ. It is clear that ρ(T¯γ,ω)=max{1-ω,ρ(T¯2,2)}. Hence, we have ρ(T¯γ,ω)<λ. For the case of λ>1, T¯2,2x2λx2 is obtained from (3.22) and equality is excluded. Hence ρ(T¯γ,ω)>λ follows from Lemma 2.8 and T¯2,2 is irreducible. Since A=(I-γL)-[(1-ω)I+(ω-γ)L+ωU] is an M-splitting of A, from Lemma 2.7, we know that λ=1 if and only if A is a singular M-matrix. So A¯=(I+S1)A is a singular M-matrix. Since A¯=(D¯-γL¯)-[(1-ω)D¯+(ω-γ)L¯+ωU¯] is an M-splitting of A¯; from Lemma 2.7 again, we have T¯γ,ω=1, which completes the proof.

In Theorem 3.4, if we let ω=γ, then can obtain some results about SOR method. For the similarity of proof of the Theorem 3.4, we only give the convergence result of the SOR method.

Theorem 3.5.

Let Tω and T¯ω be defined by (1.3) and (1.10). Assume that A is an irreducible Z-matrix and A(2:n,2:n) is an irreducible submatrix of A deleting the first row and the first column. Then for 0ω1(ω0) and ai1a1i<1, i=2,,n, we have

ρ(T¯ω)<ρ(Tω)<1 if ρ(Tω)<1;

ρ(T¯ω)=ρ(Tω)=1 if ρ(Tω)=1;

ρ(T¯ω)>ρ(Tω)>1 if ρ(Tω)>1.

4. AOR Method for <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M382"><mml:mrow><mml:mi>H</mml:mi></mml:mrow></mml:math></inline-formula>-Matrix

In this Section, we will consider AOR method for H-matrices. For convenience, we still use some notions and definitions in Section 2.

Lemma 4.1 (see [<xref ref-type="bibr" rid="B12">7</xref>]).

Let A be an H-matrix with unit diagonal elements, defining the matrices SDdiag(0,α2a2,1a1,2,,αnan,1a1,n) and SαUSD+SL+SU, where SL and SU are the strictly lower and strictly upper triangular components of SαU, respectively; then Ã=(I+Sα)A=Mα-Nα, Mα=I-SD-L-SL+Sα, and Nα=U+SU. Let u=(u1,,un)T be a positive vector such that Au>0; assume that ai10 for i=2,,n, and αi=ui-j=2i-1|ai,j|uj-j=i+1n|ai,j|uj+|ai,1|u1|ai,1|j=1n|a1,j|uj; then αi>1 for i=2,,n and for 0αi<αi, the splitting Ã=Mα-Nα is an H-splitting and ρ(Mα-1Nα)<1 so that the iteration (1.3) converges to the solution of (1.1).

Lemma 4.2.

Let A=(ai,j) be an H-matrix, and let α=min{αi},  i=2,,n, where αi is defined as Lemma 4.1. Then for any α[0,α],  Ã=(I+Sα)A is also an H-matrix.

Proof.

The conclusion is easily obtained by Lemma 4.1 .

Lemma 4.3.

Let 0γω1(ω0,γ1). Then Ã=M̃-Ñ is an H-compatible splitting.

Proof.

Let Ã=(a¯i,j) and M̃-|Ñ|=(bi,j), where M̃=(1/ω)(D̃-γL̃) and Ñ=(1/ω)[(1-ω)D̃+(ω-γ)L̃+ωŨ]. Since ãi,j={ai,j,i=1,ai,j-αiai,1a1,j,i=2,,n, we have that

if i=j, then a¯i,j=|1-αiai,1a1,i|,bi,j=1ω[|1-αiai,1a1,i|-(1-ω)|1-αiai,1a1,i|]=|1-αiai,1a1,i|;

if ij, then a¯i,j=-|ai,j-αiai,1a1,j|, since M̃-|N|=(1/ω)D̃-γL̃-(1/ω)|(1-ω)D̃+(ω-γ)L̃+ωŨ|; observe that if i<j, we have bi,j=1ω(0-ω|-ai,j+αiai,1a1,j|)=-|aij-αiai,1a1,j|. if i>j, we have bi,j=1ω[-|γ(ai,j-αiai,1a1,j)|-(ω-γ)|-ai,j+αiai,1a1,j|]=-|ai,j-αiai,1a1,j|;

Hence, we have Ã=M̃-|Ñ|, that is, Ã=M̃-Ñ is an H-compatible splitting.

Theorem 4.4.

Let the assumption of Lemma 4.2 holds. Then for any α[0,α] and 0γω1(ω0,γ1), we have ρ(T̃γ,ω)<1.

Proof.

By Lemmas 2.5, 4.2, and 4.3, the conclusion is easily obtained.

5. Numerical Examples

In this Section, we give three numerical examples to illustrate the results obtained in Sections 3 and 4.

Example 5.1.

Consider a n×n matrix of A of the form A=[1c1c2c3c1c31c1c2c1c2c3c3c11c1c2c3c2c31c1c3c1c2c31], where c1=-2/n, c2=-1/n+1, and c3=-1/n+2. It is clear that the matrix A satisfies the assumptions of Theorem 3.3. Numerical results for this matrix A are given in Table 1.

Spectral radius of the iteration matrices ρ(Tγ,ω) and ρ(T¯γ,ω) with different values of ω and γ for Example 5.1.

ωγρ(Tγ,ω)ρ(T¯γ,ω)ωγρ(Tγ,ω)ρ(T¯γ,ω)
0.40.10.99830.98400.80.70.99520.9559
0.40.40.99800.98150.80.80.99490.9529
0.50.20.99770.97900.90.70.99460.9504
0.50.40.99750.97680.90.90.99380.9431
0.60.40.99700.972210.80.99360.9411
0.60.60.99660.968910.90.99310.9367

We consider Example 5.1; if we let c1=-2/n, c2=0, and c3=-1/n+2, it is clear to show that A is an M-matrix. The initial approximation of x0 is taken as a zero vector, and b is chosen so that x=(1,2,,n)T is the solution of the linear system (1.1). Here xk+1-xk/xk+110-6 is used as the stopping criterion.

All experiments were executed on a PC using MATLAB programming package.

In order to show that the preconditioned AOR method is superior to the basic AOR method. We consider ω=γ=1, that is, the AOR method is reduced to the Gauss-Seidel method. In Table 2, we report the CPU time (T) and the number of iterations (IT) for the basic and the preconditioned Gauss-Seidel method. Here GS represents the restarted Gauss-Seidel method; the preconditioned restarted Gauss-Seidel method is noted by PGS.

CPU time and the iteration number of the basic and the preconditioned Gauss-Seidel method for Example 5.1.

nIT  (GS)CPU  (GS)IT  (PGS)CPU  (PGS)
602320.07802290.0780
903400.20303370.2030
1204460.50004430.4380
1505514.57805484.5470
1806559.59306529.5000
21075836.719075530.0470
Example 5.2.

Consider the two-dimensional convection-diffusion equation -Δu+ux+2uy=f in the unit squire Ω with Dirichlet boundary conditions see .

When the central difference scheme on a uniform grid with N×N interior nodes (N2) is applied to the discretization of the convection-diffusion equation (3.5), we can obtain a system of linear equations (1.1) of the coefficient matrix A=IP+QI, where denotes the Kronecker product, P=tridiag(-2+h8,1,-2-h8),Q=tridiag(-1+h4,1,-1-h4) are N×N tridiagonal matrices, and the step size is h=1/N.

It is clear that the matrix A is an M-matrix, so it is an H-matrix. Numerical results for this matrix A are given in Table 3.

From Table 3, for αi[0,αi), it can be seen that the convergence rate of the preconditioned Gauss-Seidel iterative method (ω=γ=1) is faster than the other preconditioned iterative method for H-matrices. And iteration numbers are not changed by the change of αi; the iteration time slightly changed by the change of αi. However, it is difficult to select the optical parameters αi and this needs a further study.

CPU time and the iteration number with various values of αi for Example 5.2.

nαi=0.5αi=0.8αi=1αi=1.2αi=2αi=0
64111111
0.06010.04880.05870.05240.05010.0629
81111111
0.05220.05040.05320.05240.05690.0635
100111111
0.05770.05470.04860.05550.05630.0663
Example 5.3.

We consider a symmetric Toeplitz matrix Tn=[abcbbabccbabbcba], where a=1, b=1/n, and c=1/n-2. It is clear that Tn is an H-matrix. The initial approximation of x0 is taken as a zero vector, and b is chosen so that x=(1,2,,n)T is the solution of the linear system (1.1). Here xk+1-xk/xk+110-6 is used as the stopping criterion see .

All experiments were executed on a PC using MATLAB programming package.

We get Table 4 by using the preconditioner P(α). We report the CPU time (T) and the number of iterations (IT) for the basic and the preconditioned AOR method. Here AOR represents the restarted AOR method; the preconditioned restarted AOR method is noted by PAOR.

CPU time and the iteration number of the basic and the preconditioned AOR method for Example 5.3.

nωγIT  (AOR)T  (AOR)IT  (PAOR)T  (PAOR)
900.90.5150.3196110.0390
1200.90.5150.1526100.0306
1800.90.5150.1407110.1096
2100.90.5150.2575110.1920
3000.90.5151.2615100.7709
4000.90.5153.2573112.3241
Remark 5.4.

In Example 5.3, we let αi>1, i=2,,n-1. From Table 4, if αi is appropriate, the convergence of the preconditioned AOR iterative method can be improved. However, it is difficult to select the optical parameters αi and this needs a further study.

Acknowledgments

The authors express their thanks to the editor Professor Paulo Batista Gonçalves and the anonymous referees who made much useful and detailed suggestions that helped them to correct some minor errors and improve the quality of the paper. This project is granted financial support from Natural Science Foundation of Shanghai (092R1408700), Shanghai Priority Academic Discipline Foundation, the Ph.D. Program Scholarship Fund of ECNU 2009, and Foundation of Zhejiang Educational Committee (Y200906482) and Ningbo Nature Science Foundation (2010A610097).

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