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., [1–14]). For example, (a) in the linear complementarity problem (LCP) (see [5, Section 10.1] for specific applications), where we are interested in finding a z∈Rn such that z≥0, Mz+q≥0, zT(Mz+q)=0, with M∈Rn×n and q∈Rn 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 [15]; (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,1≠0, 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 [18] 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 ω≠0.

To improve the convergence rate of the basic iterative methods, several preconditioned iterative methods have been proposed in [8, 12, 13, 19–24]. 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 [19] presented a modified Jacobi and Gauss-Seidel iterative methods by using the preconditioned matrix P=I+S, where P=(I+S)=[10⋯0-a211⋯0⋮⋮⋱⋮-an10⋯1].

The author [19] 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. [4] considered the generalized preconditioner used in this case is of the formP(α)=(I+Sα)=[10⋯0-α2a211⋯0⋮⋮⋱⋮-αnan10⋯1],
where α=(α2,…,αn)T∈Rn-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 x≥0(x>0), if all its entries are nonnegative (positive). Similarly, a matrix B is said to be nonnegative, denoted B≥0, if all its entries are nonnegative or, equivalently, if it leaves invariant the set of all nonnegative vectors. We compare two matrices A≥B, when A-B≥0, and two vectors x≥y(x>y) when x-y≥0(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,j≤0 for i≠j. A matrix A is called a nonsingular M-matrix if A is a Z-matrix and A-1≥0.

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|,i≠j.

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-1≥0 and N≥0

nonnegative if M-1N≥0

M-splitting if M is a nonsingular M-matrix and N≥0.

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 A≥0 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 A≥0 be a nonnegative matrix. Then the following hold.

If Ax≥βx for a vector x≥0 and x≠0, then ρ(A)≥β.

If Ax≤γx for a vector x>0, then ρ(A)≤γ; moreover, if A is irreducible and if βx≤Ax≤γx, equality excluded, for a vector x≥0 and x≠0, 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 0≤γ≤ω≤1(ω≠0,γ≠1). 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 ω≠0, γ≠1, it is easy to show that Tγ,ω is nonnegative and irreducible. By assumption, D̃,L̃,andŨ 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 ω≠0, γ≠1, 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 0≤γ≤ω≤1(ω≠0,γ≠1). 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+ωU≥0. 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 SD≥0,Sα≥0, and SL≥0. If λ<1, then from (3.11), we have T̃γ,ωx≤λx. 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,ωx≥λx 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 0≤γ≤ω≤1(ω≠0,γ≠1) 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+ωU≥0. 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¯1,2≥0 is a 1×(n-1) matrix, and T¯2,2≥0 is an (n-1)×(n-1) matrix. As A is irreducible, then at least one a1,i≠0 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 ω≠0 and γ≠1, 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 u≥0, and the first component of u is zero. Hence v≥0 and its first component is zero. Let
x=(x1x2),v=(0v2),
where x1∈R1>0, x2∈Rn-1>0, and v2∈Rn-1≥0 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)v2≠0.
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¯2,2x2≥λx2 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 0≤ω≤1(ω≠0) 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 SD≐diag(0,α2a2,1a1,2,…,αnan,1a1,n) and SαU≐SD+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 〈A〉u>0; assume that ai1≠0 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 0≤αi<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 [7].

Lemma 4.3.

Let 0≤γ≤ω≤1(ω≠0,γ≠1). 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 i≠j, 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 0≤γ≤ω≤1(ω≠0,γ≠1), 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=[1c1c2c3c1⋯c31c1c2⋱c1c2c3⋱⋱⋱c3c1⋱⋱1c1c2c3⋱c2c31c1⋮c3c1c2c31],
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.

ω

γ

ρ(Tγ,ω)

ρ(T¯γ,ω)

ω

γ

ρ(Tγ,ω)

ρ(T¯γ,ω)

0.4

0.1

0.9983

0.9840

0.8

0.7

0.9952

0.9559

0.4

0.4

0.9980

0.9815

0.8

0.8

0.9949

0.9529

0.5

0.2

0.9977

0.9790

0.9

0.7

0.9946

0.9504

0.5

0.4

0.9975

0.9768

0.9

0.9

0.9938

0.9431

0.6

0.4

0.9970

0.9722

1

0.8

0.9936

0.9411

0.6

0.6

0.9966

0.9689

1

0.9

0.9931

0.9367

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+1∥≤10-6 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.

n

IT (GS)

CPU (GS)

IT (PGS)

CPU (PGS)

60

232

0.0780

229

0.0780

90

340

0.2030

337

0.2030

120

446

0.5000

443

0.4380

150

551

4.5780

548

4.5470

180

655

9.5930

652

9.5000

210

758

36.7190

755

30.0470

Example 5.2.

Consider the two-dimensional convection-diffusion equation
-Δu+∂u∂x+2∂u∂y=f
in the unit squire Ω with Dirichlet boundary conditions see [28].

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=I⊗P+Q⊗I,
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.

n

αi=0.5

αi=0.8

αi=1

αi=1.2

αi=2

αi=0

64

1

1

1

1

1

1

0.0601

0.0488

0.0587

0.0524

0.0501

0.0629

81

1

1

1

1

1

1

0.0522

0.0504

0.0532

0.0524

0.0569

0.0635

100

1

1

1

1

1

1

0.0577

0.0547

0.0486

0.0555

0.0563

0.0663

Example 5.3.

We consider a symmetric Toeplitz matrix
Tn=[abc⋯bbab⋯ccba⋯b⋮⋮⋮⋱⋮bcb⋯a],
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+1∥≤10-6 is used as the stopping criterion see [29].

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.

n

ω

γ

IT (AOR)

T (AOR)

IT (PAOR)

T (PAOR)

90

0.9

0.5

15

0.3196

11

0.0390

120

0.9

0.5

15

0.1526

10

0.0306

180

0.9

0.5

15

0.1407

11

0.1096

210

0.9

0.5

15

0.2575

11

0.1920

300

0.9

0.5

15

1.2615

10

0.7709

400

0.9

0.5

15

3.2573

11

2.3241

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).

FrommerA.SzyldD. B.H−splittings and two-stage iterative methodsKotakemoriH.NikiH.OkamotoN.Convergence of a preconditioned iterative method for H-matricesVargaR. S.HadjidimosA.NoutsosD.TzoumasM.More on modifications and improvements of classical iterative schemes for M-matricesBermanAPlemmonsR. J.SunL.-Y.Some extensions of the improved modified Gauss-Seidel iterative method for H-matricesLiuQ.lqb_2008@hotmail.comConvergence of the modified Gauss-Seidel method for H- Matrices3Proceedings of th 3rd International Conference on Natural Computation (ICNC '07)2007Hainan, China2682712-s2.0-003740065210.1109/ICNC.2007.317LiuQ.ChenG.CaiJ.Convergence analysis of the preconditioned Gauss-Seidel method for H-matricesLiuQ.ChenG.A note on the preconditioned Gauss-Seidel method for M-matricesPooleG.BoullionT.A survey on M-matricesGalanisS.HadjidimosA.NoutsosD.On an SSOR matrix relationship and its consequencesChenX.TohK. C.PhoonK. K.cvepkk@nus.edu.sgA modified SSOR preconditioner for sparse symmetric indefinite linear systems of equationsBrussinoG.SonnadV.A comparison of direct and preconditioned iterative techniques for sparse, unsymmetric systems of linear equationsRoditisY. S.KiousisP. D.Parallel multisplitting, block Jacobi type solutions of linear systems of equationsAhnB. H.Solution of nonsymmetric linear complementarity problems by iterative methodsLiL.personal communication, 2006TsatsomerosM. J.personal communication, 2006HadjidimosA.Accelerated overrelaxation methodMilaszewiczJ. P.Improving Jacobi and Gauss-Seidel iterationsGunawardenaA. D.JainS. K.SnyderL.Modified iterative methods for consistent linear systemsKohnoT.KotakemoriH.NikiH.UsuiM.Improving the modified Gauss-Seidel method for Z-matricesEvansD. J.ShanehchiJ.Preconditioned iterative methods for the large sparse symmetric eigenvalue problemHwangF.-N.CaiX.-C.A class of parallel two-level nonlinear Schwarz preconditioned inexact Newton algorithmsBenziM.KouhiaR.TumaM.Stabilized and block approximate inverse preconditioners for problems in solid and structural mechanicsLiW.SunW.Modified Gauss-Seidel type methods and Jacobi type methods for Z-matricesYunJ. H.KimS. W.Convergence of the preconditioned AOR method for irreducible L-matricesLiY.LiC.WuS.Improving AOR method for consistent linear systemsWuM.WangL.SongY.Preconditioned AOR iterative method for linear systemsRobertM. G.