Applied Mathematics and Computation 162 (2005) 577–584

www.elsevier.com/locate/amc

How to improve MAOR methodconvergence area for linearcomplementarity problems q

Ljiljana Cvetkovi�c *, Sanja Rapaji�c

Department of Mathematics and Informatics, Faculty of Science, University of Novi Sad,

Novi Sad 21000, Serbia and Montenegro

Abstract

The linear complementarity problem can be solved by modified AOR method given

in [Appl. Math. Comput. 140 (2003) 53]. In the same paper the convergence was proved

for the H -matrix case, using the estimation of spectral radius of corresponding matrix.

In this paper we present the other possibility for obtaining convergence result. We use

the estimation of maximum norm, and surprisingly, obtain convergence area which can

be better. First, we consider SDD (strictly diagonally dominant) matrix case, and after

that H -matrix case.

� 2004 Elsevier Inc. All rights reserved.

Keywords: Linear systems; Linear complementarity problems; Iterative methods; Convergence

1. Introduction

For given matrix M 2 Rn;n and vector q 2 Rn the linear complementarity

problem LCPðM ; qÞ consists of finding a vector z 2 Rn which satisfies the

conditions

qTh

Develo*Co

E-m

0096-3

doi:10.

zP 0; Mzþ qP 0; zTðMzþ qÞ ¼ 0: ð1Þ

is work is partly supported by Republic of Serbia, Ministry of Science, Technology and

pment, grant 1840.

rresponding author.

ail addresses: lila@im.ns.ac.yu (L. Cvetkovi�c), sanja@im.ns.ac.yu (S. Rapaji�c).

003/$ - see front matter � 2004 Elsevier Inc. All rights reserved.

1016/j.amc.2003.12.108

578 L. Cvetkovi�c, S. Rapaji�c / Appl. Math. Comput. 162 (2005) 577–584

Complementarity problems can be reformulated to systems of equations in

several ways, so a large class of iterative methods has been developed forsolving these systems. We consider the LCPðM ; qÞ where M is nonsingular

matrix of the following form:

M ¼ D1 HK D2

� �; ð2Þ

D1 and D2 are square nonsingular diagonal matrices.

For solving linear system Mz ¼ b with nonsingular M-matrix and z; b 2 Rn

Young [8] proposed the modified SOR (MSOR) method, which was investi-

gated by many authors. In [5] a generalization of the MSOR method, called

MAOR method, was proposed. In [7] for matrix M having the form (2) and

strictly diagonally dominance property, among others, some sufficient condi-

tions for the convergence of the MAOR method were given. This result wasimproved in [3].

Modified AOR method for linear complementarity problems (1) with matrix

M of the form (2) is proposed in [9], with some convergence results. In this

paper we shall improve the convergence area of the MAOR method for

LCPðM ; qÞ given in [9].

We shall use the following notation. Let the diagðAÞ 2 Rn;n be the diagonal

matrix with the same diagonal elements as in given matrix A ¼ ½aij� 2 Rn;n. By

jAj and hAi we denote the absolute value of A and the comparison matrix of A,respectively. The comparison matrix hAi ¼ ½aij� 2 Rn;n is defined such that

aij ¼jaiij; i ¼ j;�jaijj; i 6¼ j:

�

Spectral radius of a matrix A is denoted by qðAÞ.

Definition 1. A matrix A ¼ ½aij� 2 Rn;n is called

ii(i) a strictly diagonally dominant (SDD) matrix if

jaiij > riðAÞ :¼Xj 6¼i

jaijj; i ¼ 1; . . . ; n;

i(ii) L-matrix if aii > 0 and aij 6 0, i 6¼ j, i; j ¼ 1; 2; . . . ; n,(iii) M-matrix if it is a nonsingular L-matrix satisfying A�1 P 0,

(iv) H -matrix if hAi is M-matrix.

Definition 2. For x 2 Rn vector xþ is defined such that ðxþÞj ¼ maxf0; xjg,j ¼ 1; 2; . . . ; n.

L. Cvetkovi�c, S. Rapaji�c / Appl. Math. Comput. 162 (2005) 577–584 579

Linear complementarity problems can be transformed to equivalent fixed-

point system of equations [1,6]. If D ¼ diagðMÞ is nonsingular matrix as in ourcase, then solving LCPðM ; qÞ is equivalent to finding a solution of the system

z ¼ ðz� D�1ðMzþ qÞÞþ:

So, in order to solve LCPðM ; qÞ a class of iterative methods for solving linearsystems has been developed.

2. MAOR method for LCPðM ; qÞ

Let M be matrix of the form (2) and let

M ¼ Dþ L1 þ U1 ð3Þ

be the standard splitting of M into diagonal (D), strictly lower (L1) and strictly

upper (U1) triangular matrices, respectively. D is supposed to be nonsingular.

Obviously,

D ¼ D1 0

0 D2

� �; L1 ¼

0 0

K 0

� �; U1 ¼

0 H0 0

� �:

Let us denote

L ¼ D�1L1 ¼0 0

D�12 K 0

� �; U ¼ D�1U1 ¼

0 D�11 H

0 0

� �:

The modified AOR (MAOR) method for LCPðM ; qÞ is defined in [9] as follows:

MAOR method

Step 1: Choose an initial approximation z0 2 Rn and set k ¼ 0,

Step 2: Calculate

zkþ1 ¼ zk�

� D�1 cL1zkþ1�

þ ðXM � cL1Þzk þ Xq��

þ; k ¼ 0; 1; . . . ; ð4Þ

Step 3: If zkþ1 ¼ zk then stop, otherwise set k :¼ k þ 1 and return to Step 2,

where X ¼ diagðx1I1;x2I2Þ, x1x2 6¼ 0, x1;x2; c 2 R and I1, I2 are the identitymatrices of the same dimensions as D1, D2, respectively.

When the parameter c equals to x2 the MAOR method reduces to the

MSOR method.

3. Convergence properties

Some sufficient conditions for the convergence of the MAOR method are

given in [9]. We shall specify the most important facts from [9] in order to

580 L. Cvetkovi�c, S. Rapaji�c / Appl. Math. Comput. 162 (2005) 577–584

explain our results. The operator f : Rn ! Rn is defined such that f ðzÞ ¼ n,where n is the fixed point of the system

n ¼ z�

� D�1ðcL1n þ ðXM � cL1Þzþ XqÞ�þ:

Denoting

Q ¼ I � cD�1L1; R ¼ I � D�1ðXM � cL1Þ

and using the next lemma proposed in [2], the convergence theorem for the

MAOR method is proved in [9].

Lemma 1 [2]. Let M be an H -matrix with positive diagonal elements. Then theLCPðM ; qÞ has a unique solution z� 2 Rn.

Theorem 1 [9]. Let M 2 Rn;n be an H -matrix of the form (2) with positivediagonal elements and J ¼ D�1ðL1 þ U1Þ. Then, for any initial guess z0 2 Rn, theiterative sequence fzkg generated by the MAOR method converges to the uniquesolution z� of the LCPðM ; qÞ and

qðhQi�1jRjÞ6 maxi¼1;2

fj1� xij þ xiqðjJ jÞg < 1;

whenever

0 < x1 <2

1þ qðjJ jÞ ; 0 < x2 <2

1þ qðjJ jÞ ; 06 c6x2:

In the proof of the previous theorem authors have obtained the following

inequality:

jzkþ1 � z�j ¼ jf ðzkÞ � f ðz�Þj6 hQi�1jRjjzk � z�j;

from which they conclude that the MAOR iterative sequence fzkg converges tothe solution z� if qðhQi�1jRjÞ < 1.

In this paper we want to expand the convergence area of the MAORmethod

given in the previous theorem. The following lemma gives us an upper bound

for khQi�1jRjk1.Let n1 and n2 be the dimensions of the diagonal matrices D1 and D2,

respectively and

N1 ¼ f1; 2; . . . ; n1g; N2 ¼ fn1 þ 1; n1 þ 2; . . . ; ng:

For all i ¼ 1; 2; . . . ; n we shall denote

li ¼ riðLÞ and ui ¼ riðUÞ:

L. Cvetkovi�c, S. Rapaji�c / Appl. Math. Comput. 162 (2005) 577–584 581

We also denote

Lx1;x2;c ¼ hQi�1jRj;‘ ¼ max

i2N2

li and u ¼ maxi2N1

ui:

Lemma 2

kLx1;x2;ck1 6 max j1f � x1j þ jx1ju; j1� x2j þ ðjcx1juþ jcjj1� x1jþ jx2 � cjÞ‘g:

Proof. From the definition of the matrix norm there exists an n-dimensionalvector y such that kyk1 ¼ 1 and

kLx1;x2;ck1 ¼ kLx1;x2;cyk1:

Let us denote Lx1;x2;cy ¼ z. Obviously,

ðI � cD�1jL1jÞz ¼ jI � X � D�1ðx2 � cÞL1 � D�1x1U1jy;

or

zi ¼ j1� x1jyi þ jx1jXk2N2

jðUÞikjyk; i 2 N1; ð5Þ

zj ¼ cXk2N1

jðLÞjkjzk þ j1� x2jyj þ jx2 � cjXk2N1

jðLÞjkjyk; j 2 N2: ð6Þ

Here we use the notation ðAÞij for the ijth element of the matrix A.From equality (5) it follows that

jzi � j1� x1jyij6 jx1jui; i 2 N1

and

jzij6 j1� x1j þ jx1ju; i 2 N1:

From the equality (6) for all j 2 N2 we have

jzj � j1� x2jyjj6Xk2N1

jðLÞjkjjczk þ jx2 � cjykj

¼Xk2N1

jðLÞjkjjczk � cj1� x1jyk þ jx2 � cjykj

6

Xk2N1

jðLÞjkjðjcjjx1juk þ jx2 � cj þ jcjj1� x1jÞ

6 ðjcjjx1juþ jx2 � cj þ jcjj1� x1jÞ‘:

582 L. Cvetkovi�c, S. Rapaji�c / Appl. Math. Comput. 162 (2005) 577–584

Finally,

jzjj6 j1� x2j þ ðjcjjx1juþ jx2 � cj þ jcjj1� x1jÞ‘; j 2 N2

and the lemma is proved. h

This lemma allows us to obtain the new convergence area of the MAOR

method for LCPðM ; qÞ, when the matrix M is strictly diagonally dominant, or

H -matrix with positive diagonal elements. First, we shall discuss the case when

M is SDD-matrix.

3.1. SDD-matrices

From now on we shall use the following notations: for x 2 R, xP 0,

tðxÞ ¼ 2

1þ x; gþðxÞ ¼ 1þ j1� xj þ xu;

f �ðxÞ ¼ 1� j1� xj � x‘‘

; f þðxÞ ¼ 1� j1� xj þ x‘‘

:

Theorem 2. Let M be an SDD-matrix with positive diagonal elements. Let ‘ andu be defined above and

t1 ¼ tðuÞ; t2 ¼ tð‘Þ:

Then the MAOR iterative method is a convergent one if the parameters x1, x2

and c are chosen in the following way

0 < x1 < t1; 0 < x2 < t2; � f �ðx2Þgþðx1Þ

< c <f þðx2Þgþðx1Þ

: ð7Þ

Proof. Since M is an SDD-matrix, it follows that ‘ < 1 and u < 1. Under this

assumption, after some algebra, it can be proved that for the above choice of

x1, x2 and c the inequalities

j1� x1j þ jx1ju < 1 and

j1� x2j þ ðjcjjx1juþ jx2 � cj þ jcjj1� x1jÞ‘ < 1

hold. Now, the convergence follows from Lemma 2. h

3.2. H -matrices

It is well known [4] that matrix M is an H -matrix if and only if there exists

the diagonal nonsingular matrix W , such that MW is SDD-matrix. This fact

allows us to prove similar convergence result for the class of H -matrices.

L. Cvetkovi�c, S. Rapaji�c / Appl. Math. Comput. 162 (2005) 577–584 583

From now on, we will suppose that we know such a matrix W , and without

loss of generalization we will suppose that W > 0.Let

~li ¼ riðW �1LW Þ and ~ui ¼ riðW �1UW Þ; i ¼ 1; 2; . . . ; n;

~‘ ¼ maxi2N2

~li and ~u ¼ maxi2N1

~ui:

Using the similar notations: t, gþ, f þ, f �, only with ~‘ and ~u instead of ‘ andu, respectively, we are getting the following result.

Theorem 3. Let M be an H -matrix with positive diagonal elements. Let ~‘ and ~ube defined above and

t1 ¼ tð~uÞ; t2 ¼ tð~‘Þ:

Then the MAOR iterative method is a convergent one if the parameters x1, x2

and c are chosen in the following way

0 < x1 < t1; 0 < x2 < t2; � f �ðx2Þgþðx1Þ

< c <f þðx2Þgþðx1Þ

: ð8Þ

Proof. If we denote by Lx1;x2;cðMW Þ the matrix which corresponds toLCPðMW ; qÞ in the same way as Lx1;x2;c corresponds to LCPðM ; qÞ, then it is

not so complicate to show that

Lx1;x2;cðMW Þ ¼ W �1Lx1;x2;cW :

Since MW is an SDD-matrix, from Theorem 2 it follows that

kLx1;x2;cðMW Þk1 < 1;

for our choice of parameters x1, x2, c. The fact that all diagonal entries of thematrix M are positive remains valid, since W is a positive matrix. Now, from

the fact that Lx1;x2;cðMW Þ and Lx1;x2;c are similar matrices it holds

qðLx1;x2;cÞ ¼ qðLx1;x2;cðMW ÞÞ6 kLx1;x2;cðMW Þk1 < 1;

which completes the proof. h

3.3. Comments and remarks

If we consider the case of H -matrices and if we want to apply Theorem 3, wehave to know scaling matrix W . This problem is not generally solvable, but

for some special subclasses of H -matrices, see [4], the construction of the

matrix W is possible.

584 L. Cvetkovi�c, S. Rapaji�c / Appl. Math. Comput. 162 (2005) 577–584

Let us restrict to case of SDD-matrices. It is obvious that convergence area

from [9] and corresponding one presented here are not subsets one of eachother. Indeed, if we try to compare expression 2

1þqðjJ jÞ with t1 ¼ 21þu and t2 ¼ 2

1þ‘,

we can see that

qðjJ jÞ6 kJk1 ¼ maxf‘; ug;

which means that the intervals for x1 and (or) x2 given in [9] can be wider than

in our case. But, the interval for c is obviously wider in our Theorem 2 than in

[9]. So, if we want to state that we have improved the convergence area in SDD

case, then we have to take the union of both areas, given here and in [9].

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