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S-SDD class of matrices and its
application
Vladimir Kostić
University of Novi Sad
Faculty of Science
Dept. of Mathematics and
Informatics
Introduction
 Equivalent definitions of S-SDD
matrices
 Bounds for the determinants
 Convergence area of PDAOR
 Subdirect sums
Equivalent definitions of
S-SDD matrices
_
S
S
ri S  A  aii






for all i  S

ri S  A rjS  A  aii  ri S  A

a jj  rjS  A

for all i  S , j  S

S
1
 A  max
iS
a
ii
 A
 ri S  A 
ri
S


min
jS
rjS  A  0
a jj  rjS  A
r
S
j
 A







S
2
L. Cvetkovic, V. Kostic and R. S. Varga, A new Gersgorin-type eigenvalue
inclusion set, Electron. Trans. Numer. Anal., 18 (2004), 73–80
 A
Equivalent definitions of
S-SDD matrices
ri S  A  aii
for all i  S
A S 
is SDD matrix
A  S 
is SDD matrix
ri S  A rjS  A 

aii  ri S  A

a jj  rjS  A
for all i  S , j  S


A is S-SDD matrix
J A  S   1S  A  , 2S  A   

Equivalent definitions of
S-SDD matrices
S
_
S






AX is an SDD
_
S
S







x





1







x  J A  S  : 1S  A  , 2S  A 

Equivalent definitions of
S-SDD matrices






H
S-SDD







1
x
x
S
1
SDD

S
2
Bounds for the determinants
det  A     aii  ri
 A 
Ostrowski 1937
det  A     aii  ui  A  
Price 1951
det  A     aii  li  A  
Ouder 1951
iN
iN
iN
k 1


det  A   max  aii   ui  A  akk
k 1..n
  max
i 1..n
i 1
li  A   ui  A 
aii
a
n
i  k 1
ii

 li  A  ,
Ostrowski 1952
Bounds for the determinants
700
600
500
400
300
200
100
0
1
2
3
4
estimates
improved estimates
determinant
Bounds for the determinants
700
600
500
400
300
200
100
0
1
2
3
4
estimates
improved estimates
determinant
Bounds for the determinants
700
600
500
400
300
200
100
0
1
2
3
4
estimates
improved estimates
determinant
Convergence of PDAOR
A is block H-matrix
n  n1  n2  ...  nN
iff
1M nisi , an
i M-matrix
1, 2,...N
H-matrix
A   A Aij 
i , j 1,2.. N
1
ii
n1 n2 n3 n4
n4
M   mij 
i , j 1,2.. N
mij   1
1 ij
1
ii
A Aij
nN



m


m

n1 1 m12 m13 m14
n2 m21 1 m23 m24
n3 m31 m32 1 m34
41
m42 m43 1
N1
mN 2 mN 3 mN 4
nN

m

m

m


1

m1N
2N
3N
4N
Convergence of PDAOR
N  
AL   Lij 
i , j 1...
AU  U ij 
i , j 1...
AD   Dij 
i , j 1...
AD  diag  D1 ,...Dn 
A  AD  AL  AU  AD
X p1   AD   AL 
1
1, 2,..., N   J1 J 2 ... J
J i     i  1   1 ,    i  1   2  ,...,   i 












1    AD      AL   AU   AD  X p    AD   AL 
1
B
Convergence of PDAOR
Let
A   Aij 
i , j 1.. N
, N  2 be block SDD matrix.
Then LPDAOR   ,  

 1 if we chose parameters in the following
way:
1 0    1, and
 2
1  ri
1  ri
   1  min
i 1.. N 2l
i 1.. N 2l
i
i
 min
1  li
1    2 min
,
i 1.. N 1  r  2l
i
i
 3 1    2 min
1   li
,
1  ri  2 li
 4
1   li
,
1  ri
i 1.. N
1    2 min
i 1.. N
and 1    min
and
2   1  ri  2li 
i 1.. N
and
or
2li
or
0    1 or
1  ri
   0.
i 1.. N 2l
i
 min
Lj.Cvetkovic, J. Obrovski,
Some convergence results of PD relaxation methods, AMC 107 (2000) 103-112
Convergence of PDAOR

lm  

k 1 sJ  k , j ) 
A1m   m A  m  k 1   s 
m   i  1   j
1
l7  A7,7
A7,1
1
 A7,7
A7,5
1
 A7,7
A7,9

1
 A7,7
A7,2


1
 A7,7
A7,6

1
 A7,7
A7,10















Convergence of PDAOR

l 
S
m
    k  1   s 
k 1 sJ  k , j ) 
S
A1m   m  A  m   k 1   s 
m   i  1   j
1
l4S  A4,4
A4,1
1
 A4,4
A4,3
1
l8S  A8,8
A8,6
1
 A8,8
A8,9

1
 A4,4
A4,2


1
 A4,4
A4,5

1
 A8,8
A8,7

1
 A8,8
A8,10

1
 A8,8
A8,11
















Convergence of PDAOR
Let
A   Aij 
Then
i , j 1.. N
, N  2 be block S-SDD matrix for set S.
LPDAOR   ,  
1
 2
 3
 4
where

 1 if we chose parameters in the following way:

0   1
 min  S ,  S     1  min  S ,  S  ,
and
1    min  S ,  S 
and
0    1,
1    min  S    ,  S   
and
 min  S ,  S     0.


or
or
1  l 1  r   l r
  2 min
,
1

r

2
l
1

r

r

2
l
r




 

1   l 1  r    l r
1   l 1  r    l r
,      2 min
    2 min
1  r 1  r   r r
1  r  2 l 1  r    r  2 l  r
 2   1  r  2l   1  r    r  2l  r .
1
   min
2
l 1  r   l r
S
i
S
i , jS
S
S
j
S
i
S
j
S
S
i
S
S
i
S
i
S
i
i
S
j
S
j
S
i
S
j
S
j
S
i
S
S
j
S
i
S
j
S
S
i
i
S
j
S
S
i
i
i , jS
i , jS
i
S
i
i
S
1    min  S   ,  S   ,
and
1    min  S    ,  S   
1  ri S 1  rjS  ri S rjS
1
 S  min
,
2 i , jS
liS 1  rjS  liS rjS
S
or
S
i
S
j
S
S
j
i , jS
i
S
i
S
j
S
j
S
j
Cvetković, Kostic, New subclasses of block H-matrices with applications to
parallel decomposition-type relaxation methods, Numer. Alg. 42 (2006)
S
i
S
i
S
j
S
j
,
Convergence of PDAOR
Subdirect sums
A
n1 n1
C  A k B
B
n2 n2
Subdirect sums
 B11 B12 
 A11 A12 
A22 , B11 
B
A


 B21 B22  n2 n2
 A21 A22  n1n1
Same sign
pattern on
the diagonal
A12
0
 A11


C   A21 A22  B11 B12 
 0
B21
B22   n  n k  n  n k 
1
2
1
2
k k
Subdirect sums
H
NO
A and B are
is the matrix C
YES
SDD
,
too?
Subdirect sums
H
NO
A and B are
is the matrix C
,
too?
S-SDD
YES
SDD
R. Bru, F. Pedroche, and D. B. Szyld, Subdirect sums of S-Strictly Diagonally
Dominant matrices, Electron. J. Linear Algebra, 15 (2006), 201–209
Subdirect sums












Subdirect sums
S













aii  ri S  A 

a jj  rjS  A 

ri S  A  rjS  A 
for all i  S , j  S

Subdirect sums
Let S  1, 2,..., card  S   n1  k
If
A is S-SDD and B is SDD then
C  A k B
is
S-SDD.
R. Bru, F. Pedroche, and D. B. Szyld, Subdirect sums of S-Strictly Diagonally
Dominant matrices, Electron. J. Linear Algebra, 15 (2006), 201–209
Subdirect sums












S
Subdirect sums of S-SDD matrices












SA
Subdirect sums of S-SDD matrices












SB
Subdirect sums of S-SDD matrices
Let
If
S be arbitrary
A is S A - SDD, B is S B - SDD
and
J A  S A   J B  SB    then
C  A k B is S-SDD.
Thank you for your attention