Int. J.IndustrialMathematisVol. 2,No. 3(2010)181-187
Two Modied Jaobi Methods for M-Matries
Z.Lorkojori a
,N.Mikaeilvand b
(a)Department ofMathematis,IslamiAzadUniversity,Ghaemshahrbranh, Ghaemshahr,Iran.
(b)Department ofMathematis,IslamiAzadUniversity,Ardabilbranh,Ardabil,Iran.
Reeived19June2010;revised 8November 2010;aepted13November2010.
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In 2009 B. Zheng et al. proposed two modied Gauss-Seidel (MGS) methods for linear
systemwithM-matries. Inthispaper,weusethepreonditionersintroduedbyB.Zheng
et al . For modied Jaobi method. The omparison theorems and numerial examples
show thattheproposedmethodsaresuperiorto the lassialJaobimethod.
Keywords: Iterativemethods; M-matrix;Preonditioning;Jaobimethod.
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1 Introdution
We onsider thefollowingpreonditionedlinear system
PAX =Pb (1.1)
where A = (a
ij ) 2 R
nn
is an M-matrix, P 2 R nn
is a preonditioner and X;b 2 R n
are vetors. Without loss of generality, we assume that A has a splitting of the form
A=I L U, whereI istheidentitymatrix, Land U arestritlylowerand stritly
uppertriangularpartsof A,respetively.
The preonditionerP
Smax
wasintroduedbyKotakemori et al. [4℄ asfollows:
P
Smax
=I+S
max
(1.2)
whereS
max
isdenedby
S
max =(S
m
ij )=
a
i;k
i
i=1:::;n 1;j >i;
o; Otherwise;
(1.3)
k
i
=minfjjmaxj a
ij
j;i<ng:
In 2009 Zhenget al. [19 ℄ proposedthefollowingtwo preonditioners:
P
max
=I+S
max +R
max
(1.4)
and
P
R
=I +S
max
+R ; (1.5)
where
(R
max )
i;j =
a
n;k
n
i=n;j=k
n ;
o; OtherWise
(1.6)
withk
n
=minfjjja
n;j
j=maxfja
n;l
j;l=1;:::;n 1gg and
R
i;j =
a
i;j
; i=n;1j n 1;
0; OtherWise:
(1.7)
Forthepreonditioner(1.4), thepreonditionedmatrixA
max
=(I+S
max +R
max )Aan
besplit asfollows:
A
max
= M
max N
max
(1.8)
= (I D
D) (L R
max
+E+
E+U S
max
+F +S
max
U) (1.9)
whereD; EandF arerespetivelythediagonal,stritlylowerandstritlyuppertriangular
partsofS
max
L;while
Dand
Earethediagonal,stritlylowertriangularpartsofR
max (L+
U). If M
max
isnonsingular,themodiedJaobiiterative matrixisdened by:
T
max =M
1
max N
max
=(I D
D) 1
(L R
max
+E+
E+U S
max
+F +S
max U)
Forthepreonditoner(1:5),thepreonditionedmatrixA
R
=(I+S
max
+R )Aanbesplit
as
A
R =M
R N
R
=(I D
D) (L R+E+
E+U S
max
+F +S
max
U); (1.10)
where
D and
E are the diagonal, stritly lower triangular parts of R (L+U). If M
R is
nonsingular,themodied Jaobiiterative matrixisdened by
T
R =M
1
R N
R
=(I D
D) 1
(L R+E+
E+U S
max
+F +S
max U):
Thispaperisorganizedasfollows. Insetion2,wepresentsomenotations,denitionsand
preliminaryresults. In setion3, we prove the onvergeneof theproposedmethodsand
some omparison theorems. In setion 4 we present some numerial examples to onrm
ourtheoretial analysis. Finally, inSetion5,onlusion isdrown.
2 Preliminaries
For A = (a
i;j
);B = (b
i;j ) 2 R
nn
; we write A B if a
i;j b
i;j
holds for all i;j =
1;2;:::;n: Calling A nonnegative if A 0(a
i;j
0i;j = 1;:::;n), where 0 is an nn
zero matrix. Forthe vetors a;b 2R n1
;a b and a 0 an be dened in the similar
manner.
Denition 2.1. A matrix A is L-matrix if a
i;i
> 0;i = 1;:::;n and a
i;j
0 for all
Lemma 2.1. [13 ℄Let A be a nonnegative nn nonzero matrix. Then
1. (A); the spetral radius of A, isan eigenvalue;
2. A has a nonnegative eigenvetor orresponding to (A);
3. (A) is a simple eigenvalue of A;
4. (A) inreases when any entry of A inreases.
Denition 2.2. Let A be a real matrix. Then
A=M N
is alled a splitting of A if M is a nonsingular matrix. Thesplitting isalled
1. weak regular ifM is nonsingular, M 1
0 and M 1
N 0;
2. regular if M isnonsingular, M 1
0 and N 0;
3. nonnegative if M 1
N 0;
4. M-splitting if M isa nonsingular M-matrix and N 0.
Denition 2.3. We all A=M N the Jaobi splitting of A, if M =I is nonsingular
and N =(L+U). In addition, the splitting is alled
1. Jaobi onvergent if (M 1
N)<1;
2. Jaobi regular if M 1
=I 1
0 and N =(L+U)0.
Lemma2.2. [7℄ LetA=M N bean M-splitting of A. Then (M 1
N)<1 if andif A
is a nonsingular M-matrix.
Lemma 2.3. [15 ℄ Let A be a nonsingular M-matrix, and let A =M
1 N
1 =M
2 N
2
be two onvergent splittings, the rst one weak regular and the seond one regular. if
M 1
1
M
1
2 then
(M 1
1 N
1
)(M 1
2 N
2 )<1:
3 Convergene And Comparison Theorems
Before proving the omparison theorems, we prove the onvergene of modied Jaobi
method with the preonditioner P
Smax
= I+S
max
, the preonditioned matrix A
Smax =
(I +S
max
)Aan bewrittenas
A
Smax =M
Smax N
Smax
=(I D) (L+E+U S
max
+F +S
max U):
In whih D;E and F are dened as insetion 1. Hene, ifa
i;k
i a
k
i ;i
6=1(i =1;2;;n 1)
thenthe modiedJaobiiterative matrixT
Smax
an be denedby
T
Smax =M
1
Smax N
Smax
=(I D) 1
(L+E+U S
max
+F +S
max U)
Lemma3.1. LetA=I L U beanonsingular M-matrix. Assumethat 0a i;k i a k i ;i <
1;1in 1, then A
S max =M S max N S max
is regular and Jaobi onvergent.
Proof: The elements of A
Smax are a m i;j = a i;j a i;k i a k i ;j
. We observe that when
0 a
i;ki a
ki;i
<1;1 in 1,the diagonal elements of A
Smax
are positive and M 1
S
max
exists. It is known that (see [1 ℄) an L-matrix A is a nonsingular M-matrix if and only
if there exists a positive vetor y suh that Ay > 0. By taking suh y, the fat that
I+S
max
0 impliesA
Smax
y=(I+S
max
)Ay>0.
Consequently,theL-matrixA
Smax
isanonsingularM-matrix,whihmeansA 1
Smax 0.
Sine 0a
i;k i a k i ;i
<1,we have (I D) 1
I, thefollowing inequalityholds:
M 1
Smax
=(I D) 1
0
sine U S
max
0 learly N
S
max
0 holds. Therefore, A
S max = M S max N S max is a
regular andJaobionvergent splittingbydenition(2.3) andlemma (2.2).
3.1 On The Preonditioner R
max
=I+S
max +R
max
Theorem 3.1. Let A be a nonsingular M-matrix and let that 0 a
i;k i a k i ;i
<1;1 i
n 1 and 0 a
i;k j a k j ;n
<1;k
j
= 1; ;n 1; then A
max =M
max N
max
is a regular
and Jaobi onvergent splitting.
Proof: We observe that when 0 a
i;k i a k i ;i
<1;1 in 1 and 0 a
n;k j a k j ;n < 1;k j
=1; ;n 1thediagonalelementsofA
max
arepositiveandM 1
max
exists. Similarto
theproof oflemma(3.1), we an showthatA
max
=(I+S
max +R
max
)Aisa nonsingular
M-matrix whenA is anonsingular M-matrix.
Thus A 1
max
0. When 0 a
i;k i a k i ;i
< 1;1 i n 1 and 0 a
n;k j a k j ;n
<1;k
j =
1; ;n 1,wehave D+
D <I so(I D
D)0 thefollowingrelationholds:
M 1
max
=(I D
D) 1
=(I (D+
D)) 1
=fI+(D+
D)+(D+
D) 2
++(D+
D) n 1
g0
SineLR
max
0andU S
max
0,learlyN
max
=L R
max +E+
E+U S
max +
F +S
max
U 0. Then A
max =M
max N
max
isaregular andJaobionvergent splitting
bydenition(2.3) and lemma (2.2).
Theorem3.2. LetAbeanonsingularM-matrix. Thenundertheassumptionsoftheorem
(3.1), the following relation holds,
(T
max
)(T)<1:
Proof: The iteration matrix of the lassial Jaobi method for A is T = (L +U).
SineAisanonsingularM-matrix,thelassialJaobisplittingA=I (L+U)islearly
regularand onvergent. Fromtheorem (3.1), A
max =P
max
A=M
max N
max
is aJaobi
onvergent splitting. To ompare (T
max
) with (T) , we onsider thefollowing splitting
of A:
A
max =P
max
A=(I+S
max +R
max
)A=M
max N
max
and hene,
A=(I+S +R ) 1
M (I+S +R )
1
If we take M
1
=(I +S
max +R
max )
1
M
max
and N
1
=(I +S
max +R
max )
1
N
max , then
(M 1
1 N
1
)<1 sineM 1
max N
max =M
1
1 N
1
. Also,note that
M 1
1
=M 1
max (I+S
max +R
max
)=(I D
D) 1
(I+S
max +R
max
)(I D
D) 1
I 1
:
If follows from lemma (2.3) that (M 1
1 N
1
) < (M 1
N) < 1. Hene (M 1
max N
max ) <
(M 1
N)<1,i.e., (T
max
)(T)<1.
3.2 On The Preonditioner R
R
=I+S
max +R
Theorem 3.3. Let A be a nonsingular M-matrix and let that 0 a
i;k
i a
k
i ;i
<1;1 i
n 1 and 0 P
n 1
k=1 a
n;k a
k;n
<1; then A
R =M
R N
R
isregular and Jaobi onvergent
splitting.
Proof: Theproof issame as theproof oftheorem (3.1).
Similar totheprooftheorem (3.2),we an ompare(T) with(T
R
). Thefollowingis
a omparisonresult andwe willstate itwithoutproof.
Theorem3.4. LetAbeanonsingularM-matrix. Thenundertheassumptionsoftheorem
3:3, the following relation holds,
(T
R
)(T)<1:
4 Examples
In thissetion, we test thefollowingmatrix,
A= 0
B
B
1:00 0:00 0:20 0:60
0:10 1:00 0:10 0:50
0:30 0:10 1:00 0:10
0:40 0:30 0:10 1:00 1
C
C
A
By using preonditioners I +S
max +R
max
and I +S
max
+R , we have the following
matries :
A
max =
0
B
B
0:76 0:18 0:26 0:00
0:30 0:85 0:15 0:00
0:34 0:13 0:99 0:00
0:00 0:30 0:18 0:76 1
C
C
A
and
A
R =
0
B
B
0:76 0:18 0:26 0:00
0:30 0:85 0:15 0:00
0:34 0:13 0:99 0:00
0:06 0:01 0:11 0:60 1
C
C
A
By omputation, we have
(M 1
N)=0:736125>(M 1
max N
max
)=0:530363
and
(M 1
N)=0:736125>(M 1
N
R
Next,we test thefollowingmatrix:
A= 0
B
B
B
B
B
B
B
B
B
B
B
B
B
B
1:0 0:1 0:0 0:1 0:0 0:1 0:0 0:2 0:4 0:0
0:1 1:0 0:1 0:3 0:1 0:0 0:0 0:1 0:1 0:0
0:2 0:1 1:0 0:1 0:0 0:1 0:0 0:0 0:3 0:1
0:1 0:1 0:0 1:0 0:0 0:1 0:4 0:0 0:1 0:0
0:0 0:1 0:0 0:1 1:0 0:4 0:2 0:0 0:1 0:1
0:2 0:0 0:1 0:0 0:0 1:0 0:0 0:4 0:1 0:1
0:0 0:1 0:2 0:1 0:0 0:1 1:0 0:0 0:3 0:1
0:2 0:1 0:2 0:0 0:0 0:1 0:0 1:0 0:3 0:0
0:0 0:1 0:1 0:0 0:2 0:0 0:1 0:2 1:0 0:1
0:1 0:0 0:0 0:1 0:0 0:1 0:3 0:0 0:1 1:0 1
C
C
C
C
C
C
C
C
C
C
C
C
C
C
A
We have
(T)=0:856049;(T
max
)=0:774723 and (T
R
)=0:774471. Clearly, (T
max
)<(T)
and (T
R
)<(T)holds.
5 Conlusion
In 1991, A. D. Gunawardena et al. proposed the modied Gauss-Siedel (MGS) method
for solving the linear system with the preonditioned P = I +S [A. D. Gunawardena,
S. K. Jain, L. Snider, Modied Iterative Method For Consistent Linear System, Linear
Algebra Appl. 154-156 (1991)123-143℄. Based on theirwork, in2009 B. Zhenget al.[19 ℄
proposed two modied Gauss-Seidel (MGS) methods for linear system with M-matries.
In this paper, we used the preonditioners introdued by B. Zheng et al . for modied
Jaobimethod. Also,the omparisontheoremsand numerialexampleswereshown that
theproposedmethodsaresuperiorto thelassialJaobimethod.
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