Directory UMM :Data Elmu:jurnal:J-a:Journal of Computational And Applied Mathematics:Vol103.Issue2:
Journal of Computational and Applied Mathematics 103 (1999) 281–285
A note on the convergence of the MAOR method
Ljiljana Cvetkovic ∗
Faculty of Science, Institute of Mathematics, Trg. D. Obradovica 4, 21000 Novi Sad, Yugoslavia
Received 27 October 1997; received in revised form 12 October 1998
Abstract
The MAOR method as a generalization of the well-known MSOR method was introduced by Hadjidimos et al. (Appl.
Numer. Math. 10 (1992) 115–127) and investigated in Y. Song (J. Comput. Appl. Math. 79 (1997) 299–317) where some
convergence results for the case when matrix of the system is strictly diagonally dominant are obtained. In this paper we
c 1999 Elsevier Science B.V. All rights reserved.
shall improve these results.
AMS classi cation: 65F10
Keywords: Linear systems; Iterative methods; Convergence
1. Introduction
Let
Ax = b
(1)
be a system of linear equations with the n × n nonsingular matrix A of the following form:
A=
"
D1
−H
−K
D2
#
;
(2)
where D1 and D2 are square nonsingular diagonal matrices.
For solving system (1) Young [4] proposed the modi ed SOR (MSOR) method, which was
investigated by many authors (for detailed comments see [2, 4]).
∗
E-mail: [email protected].
c 1999 Elsevier Science B.V. All rights reserved.
0377-0427/99/$ – see front matter
PII: S 0 3 7 7 - 0 4 2 7 ( 9 8 ) 0 0 2 6 3 - 5
282
L. Cvetkovic / Journal of Computational and Applied Mathematics 103 (1999) 281–285
In [1] a generalization of the MSOR method, called MAOR method, was proposed. In [2] for
matrix A having the form (2) and strictly diagonally dominance property, among others, some
sucient conditions for the convergence of the MAOR method were given. Here we shall improve
this result.
De nition 1. Matrix A ∈ C n;n is a strictly diagonally dominant (SDD) matrix if
|aii |¿ri (A) :=
X
|aij |;
i = 1; : : : ; n:
j6=i
2. MAOR method
Let A be two-cyclic matrix of the form (2) and let
A = D−CL −CU
be the standard splitting of A into diagonal (D), strictly lower (CL ) and strictly upper (CU ) triangular
matrices, respectively. D is suppose to be nonsingular. Obviously,
D=
"
D1
0
0
D2
#
"
CL =
;
0
0
K
0
#
CU =
;
"
0
H
0
0
#
:
Let us denote
−1
L = D CL =
"
0
0
D2−1 K
0
#
;
−1
U = D CU =
"
0
D1−1 H
0
0
#
:
The modi ed AOR (MAOR) method is de ned as follows:
x k+1 = L!1 ; !2 ;
x k + !1 ; !2 ;
; b ;
k = 0; 1; : : : ;
(3)
where
L!1 ; !2 ;
= (I −
L)−1 (I −
+ (!2 −
)L + !1 U );
!1 ; !2 ;
;b = (I −
L)−1 D−1
b;
and
= diag(!1 I1 ; !2 I2 ); !1 !2 6= 0; !1 ; !2 ;
∈ R and I1 ; I2 are the identity matrices of the same dimensions as D1 ; D2 , respectively.
When the parameter
equals !2 the MAOR method reduces to the MSOR method.
3. Convergence properties
It is well known that MAOR method will converge for any start vector if and only if (L!1 ;!2 ;
)¡1.
The following Lemma gives us an upper bound for kL!1 ; ! 2 ;
k∞ .
L. Cvetkovic / Journal of Computational and Applied Mathematics 103 (1999) 281–285
283
Let n1 and n2 be the dimensions of the diagonal matrices D1 and D2 , respectively and
N1 = {1; 2; : : : ; n1 };
N2 = {n1 + 1; n1 + 2; : : : ; n}:
For all i = 1; 2; : : : ; n we shall denote
li = ri (L) and
ui = ri (U ):
We also denote
‘ = max li
i∈N2
and
u = max ui :
i∈N1
Lemma 2.
kL!1 ; !2 ;
k∞ 6 max{|1−!1 | + |!1 |u; |1−!2 | + (|!2 −
!1 | + |
!1 |u)‘}:
Proof. There exists an n-dimensional vector y such that kyk∞ = 1 and
kL!1 ; ! 2 ;
k∞ = kL!1 ;! 2 ;
yk∞ :
Let us denote L!1 ; ! 2 ;
y = z: Obviously,
(I −
L)z = (I −
+ (!2 −
)L + !1 U )y
or
zi = (1−!1 )yi + !1
X
(U )ik yk ;
i ∈ N1 ;
(4)
k∈N2
zj =
X
(L)jk zk + (1−!2 )yj + (!2 −
)
X
(L)jk yk ;
j ∈ N2 :
k∈N 1
k∈N 1
Here we use the notation (A)ij for the ijth element of the matrix A.
From the equality (4) it follows that
|zi −(1−!1 )yi |6|!1 |ui ;
i ∈ N1
and
|zi |6|1−!1 | + |!1 |u:
From the second equality (5) for all j ∈ N2 we have
|zj −(1−!2 )yj | 6
X
|(L)jk ||
zk + (!2 −
)yk |
X
|(L)jk ||
(zk −(1−!1 )yk ) + (!2 −
!1 )yk |
X
|(L)jk |(|
||!1 |uk + |!2 −
!1 |)
k∈N 1
=
k∈N 1
6
k∈N 1
6 (|
||!1 |u + |!2 −
!1 |)‘:
(5)
284
L. Cvetkovic / Journal of Computational and Applied Mathematics 103 (1999) 281–285
Finally,
|zj |6|1−!2 | + (|
||!1 |u + |!2 −
!1 |)‘
and the lemma is proved.
This Lemma allows us to obtain the convergence area for MAOR method when the matrix of our
system is strictly diagonally dominant.
3.1. SDD matrices
From now on we shall use the following notations: for x ∈ R; x¿0;
t(x) =
2
;
1+x
f− (x) =
g− (x) = −f− (x)
1−|1−x|−x‘
;
‘(1 + u)
f+ (x) =
1−|1−x| + x‘
;
‘(1 + u)
1+u
:
1−u
Theorem 3. Let A be an SDD matrix. Let ‘ and u are as de ned above and
t1 = t(u);
t2 = t(‘);
t3 = t(‘u):
Then the MAOR iterative method is a convergent one if the parameters !1 ; !2 and
are chosen
in the following way:
0¡!1 ¡t1 ; 0¡!2 ¡t2 ;
−f− (!2 )¡
!1 ¡f+ (!2 )
(6)
0¡!1 ¡t1 ; t2 6!2 ¡t3 ;
g− (!2 )¡
!1 ¡f+ (!2 ):
(7)
or
Proof. Since A is an SDD matrix, it follows that ‘¡1 and u¡1. Under this assumption it is easy
to prove that for all above choices of !1 , !2 and
the inequalities
|1−!1 | + |!1 |u¡1
and
|1−!2 | + (|!2 −
!1 | + |
!1 |u)‘¡1
hold. Now, the convergence follows from Lemma 2.
Comments and remarks. We shall continue to use the notations ‘ and u which correspond to 2
and 1 from [2], respectively.
Let us compare our result with the Theorem 3.6 from [2]. The convergence area for parameters
!1 ; !2 and
in Theorem 3.6 from [2] is
0¡!1 ¡t1 ;
0¡!2 ¡t2 ;
06
6!2
(8)
L. Cvetkovic / Journal of Computational and Applied Mathematics 103 (1999) 281–285
285
or
0¡!2 6
¡t2 ;
0¡
!1 ¡!2 t1 :
(9)
If (8) is satis ed, then we have
06
!1 6
2!2
= !2 t1 :
1+u
Since
+
!2 t1 ¡f (!2 ) =
!2 + ! 2 ‘
‘(1 + u)
2−!2 + !2 ‘
‘(1 + u)
for 0¡!2 61;
for 1¡!2 ¡t2 ;
we conclude that for such a choice of parameters (6) is satis ed.
In the second case, if (9) is satis ed, similarly we conclude that
0¡
!1 ¡f+ (!2 ):
Since 0¡!2 ¡t2 and
0¡!1 ¡
2 !2
2
6
= t1 ;
1+u
1+u
we have again that (6) is satis ed. Hence, the convergence area from our Theorem 3 is always
wider than the corresponding one in Theorem 3.6 from [2].
In the case of MSOR method, when
= !2 our Theorem 3 states that the MSOR method will
converge if
0¡!1 ¡t1 ;
0¡!2 ¡t2
or
1 + ‘−(2=!2 )
(2=!2 )−1 + ‘
¡!1 ¡
;
‘(1−u)
‘(1 + u)
t2 6!2 ¡t3 ;
which is obviously wider than the corresponding one in Theorem 3.6 from [2].
References
[1] A. Hadjidimos, A. Psimarni, A.K. Yeyios, On the convergence of the modi ed accelerated overrelaxation (MAOR)
method, Appl. Numer. Math. 10 (1992) 115–127.
[2] Y. Song, On the convergence of the MAOR method, J. Comput. Appl. Math. 79 (1997) 299–317.
[3] R.S. Varga, Matrix Iterative Analysis, Prentice-Hall, Englewood Cli s, NJ, 1962.
[4] D.M. Young, Iterative Solution of Large Linear Systems, Academic Press, New York, 1971.
A note on the convergence of the MAOR method
Ljiljana Cvetkovic ∗
Faculty of Science, Institute of Mathematics, Trg. D. Obradovica 4, 21000 Novi Sad, Yugoslavia
Received 27 October 1997; received in revised form 12 October 1998
Abstract
The MAOR method as a generalization of the well-known MSOR method was introduced by Hadjidimos et al. (Appl.
Numer. Math. 10 (1992) 115–127) and investigated in Y. Song (J. Comput. Appl. Math. 79 (1997) 299–317) where some
convergence results for the case when matrix of the system is strictly diagonally dominant are obtained. In this paper we
c 1999 Elsevier Science B.V. All rights reserved.
shall improve these results.
AMS classi cation: 65F10
Keywords: Linear systems; Iterative methods; Convergence
1. Introduction
Let
Ax = b
(1)
be a system of linear equations with the n × n nonsingular matrix A of the following form:
A=
"
D1
−H
−K
D2
#
;
(2)
where D1 and D2 are square nonsingular diagonal matrices.
For solving system (1) Young [4] proposed the modi ed SOR (MSOR) method, which was
investigated by many authors (for detailed comments see [2, 4]).
∗
E-mail: [email protected].
c 1999 Elsevier Science B.V. All rights reserved.
0377-0427/99/$ – see front matter
PII: S 0 3 7 7 - 0 4 2 7 ( 9 8 ) 0 0 2 6 3 - 5
282
L. Cvetkovic / Journal of Computational and Applied Mathematics 103 (1999) 281–285
In [1] a generalization of the MSOR method, called MAOR method, was proposed. In [2] for
matrix A having the form (2) and strictly diagonally dominance property, among others, some
sucient conditions for the convergence of the MAOR method were given. Here we shall improve
this result.
De nition 1. Matrix A ∈ C n;n is a strictly diagonally dominant (SDD) matrix if
|aii |¿ri (A) :=
X
|aij |;
i = 1; : : : ; n:
j6=i
2. MAOR method
Let A be two-cyclic matrix of the form (2) and let
A = D−CL −CU
be the standard splitting of A into diagonal (D), strictly lower (CL ) and strictly upper (CU ) triangular
matrices, respectively. D is suppose to be nonsingular. Obviously,
D=
"
D1
0
0
D2
#
"
CL =
;
0
0
K
0
#
CU =
;
"
0
H
0
0
#
:
Let us denote
−1
L = D CL =
"
0
0
D2−1 K
0
#
;
−1
U = D CU =
"
0
D1−1 H
0
0
#
:
The modi ed AOR (MAOR) method is de ned as follows:
x k+1 = L!1 ; !2 ;
x k + !1 ; !2 ;
; b ;
k = 0; 1; : : : ;
(3)
where
L!1 ; !2 ;
= (I −
L)−1 (I −
+ (!2 −
)L + !1 U );
!1 ; !2 ;
;b = (I −
L)−1 D−1
b;
and
= diag(!1 I1 ; !2 I2 ); !1 !2 6= 0; !1 ; !2 ;
∈ R and I1 ; I2 are the identity matrices of the same dimensions as D1 ; D2 , respectively.
When the parameter
equals !2 the MAOR method reduces to the MSOR method.
3. Convergence properties
It is well known that MAOR method will converge for any start vector if and only if (L!1 ;!2 ;
)¡1.
The following Lemma gives us an upper bound for kL!1 ; ! 2 ;
k∞ .
L. Cvetkovic / Journal of Computational and Applied Mathematics 103 (1999) 281–285
283
Let n1 and n2 be the dimensions of the diagonal matrices D1 and D2 , respectively and
N1 = {1; 2; : : : ; n1 };
N2 = {n1 + 1; n1 + 2; : : : ; n}:
For all i = 1; 2; : : : ; n we shall denote
li = ri (L) and
ui = ri (U ):
We also denote
‘ = max li
i∈N2
and
u = max ui :
i∈N1
Lemma 2.
kL!1 ; !2 ;
k∞ 6 max{|1−!1 | + |!1 |u; |1−!2 | + (|!2 −
!1 | + |
!1 |u)‘}:
Proof. There exists an n-dimensional vector y such that kyk∞ = 1 and
kL!1 ; ! 2 ;
k∞ = kL!1 ;! 2 ;
yk∞ :
Let us denote L!1 ; ! 2 ;
y = z: Obviously,
(I −
L)z = (I −
+ (!2 −
)L + !1 U )y
or
zi = (1−!1 )yi + !1
X
(U )ik yk ;
i ∈ N1 ;
(4)
k∈N2
zj =
X
(L)jk zk + (1−!2 )yj + (!2 −
)
X
(L)jk yk ;
j ∈ N2 :
k∈N 1
k∈N 1
Here we use the notation (A)ij for the ijth element of the matrix A.
From the equality (4) it follows that
|zi −(1−!1 )yi |6|!1 |ui ;
i ∈ N1
and
|zi |6|1−!1 | + |!1 |u:
From the second equality (5) for all j ∈ N2 we have
|zj −(1−!2 )yj | 6
X
|(L)jk ||
zk + (!2 −
)yk |
X
|(L)jk ||
(zk −(1−!1 )yk ) + (!2 −
!1 )yk |
X
|(L)jk |(|
||!1 |uk + |!2 −
!1 |)
k∈N 1
=
k∈N 1
6
k∈N 1
6 (|
||!1 |u + |!2 −
!1 |)‘:
(5)
284
L. Cvetkovic / Journal of Computational and Applied Mathematics 103 (1999) 281–285
Finally,
|zj |6|1−!2 | + (|
||!1 |u + |!2 −
!1 |)‘
and the lemma is proved.
This Lemma allows us to obtain the convergence area for MAOR method when the matrix of our
system is strictly diagonally dominant.
3.1. SDD matrices
From now on we shall use the following notations: for x ∈ R; x¿0;
t(x) =
2
;
1+x
f− (x) =
g− (x) = −f− (x)
1−|1−x|−x‘
;
‘(1 + u)
f+ (x) =
1−|1−x| + x‘
;
‘(1 + u)
1+u
:
1−u
Theorem 3. Let A be an SDD matrix. Let ‘ and u are as de ned above and
t1 = t(u);
t2 = t(‘);
t3 = t(‘u):
Then the MAOR iterative method is a convergent one if the parameters !1 ; !2 and
are chosen
in the following way:
0¡!1 ¡t1 ; 0¡!2 ¡t2 ;
−f− (!2 )¡
!1 ¡f+ (!2 )
(6)
0¡!1 ¡t1 ; t2 6!2 ¡t3 ;
g− (!2 )¡
!1 ¡f+ (!2 ):
(7)
or
Proof. Since A is an SDD matrix, it follows that ‘¡1 and u¡1. Under this assumption it is easy
to prove that for all above choices of !1 , !2 and
the inequalities
|1−!1 | + |!1 |u¡1
and
|1−!2 | + (|!2 −
!1 | + |
!1 |u)‘¡1
hold. Now, the convergence follows from Lemma 2.
Comments and remarks. We shall continue to use the notations ‘ and u which correspond to 2
and 1 from [2], respectively.
Let us compare our result with the Theorem 3.6 from [2]. The convergence area for parameters
!1 ; !2 and
in Theorem 3.6 from [2] is
0¡!1 ¡t1 ;
0¡!2 ¡t2 ;
06
6!2
(8)
L. Cvetkovic / Journal of Computational and Applied Mathematics 103 (1999) 281–285
285
or
0¡!2 6
¡t2 ;
0¡
!1 ¡!2 t1 :
(9)
If (8) is satis ed, then we have
06
!1 6
2!2
= !2 t1 :
1+u
Since
+
!2 t1 ¡f (!2 ) =
!2 + ! 2 ‘
‘(1 + u)
2−!2 + !2 ‘
‘(1 + u)
for 0¡!2 61;
for 1¡!2 ¡t2 ;
we conclude that for such a choice of parameters (6) is satis ed.
In the second case, if (9) is satis ed, similarly we conclude that
0¡
!1 ¡f+ (!2 ):
Since 0¡!2 ¡t2 and
0¡!1 ¡
2 !2
2
6
= t1 ;
1+u
1+u
we have again that (6) is satis ed. Hence, the convergence area from our Theorem 3 is always
wider than the corresponding one in Theorem 3.6 from [2].
In the case of MSOR method, when
= !2 our Theorem 3 states that the MSOR method will
converge if
0¡!1 ¡t1 ;
0¡!2 ¡t2
or
1 + ‘−(2=!2 )
(2=!2 )−1 + ‘
¡!1 ¡
;
‘(1−u)
‘(1 + u)
t2 6!2 ¡t3 ;
which is obviously wider than the corresponding one in Theorem 3.6 from [2].
References
[1] A. Hadjidimos, A. Psimarni, A.K. Yeyios, On the convergence of the modi ed accelerated overrelaxation (MAOR)
method, Appl. Numer. Math. 10 (1992) 115–127.
[2] Y. Song, On the convergence of the MAOR method, J. Comput. Appl. Math. 79 (1997) 299–317.
[3] R.S. Varga, Matrix Iterative Analysis, Prentice-Hall, Englewood Cli s, NJ, 1962.
[4] D.M. Young, Iterative Solution of Large Linear Systems, Academic Press, New York, 1971.