S-SDD class of matrices and its S-SDD class of matrices and its applicationapplication
Vladimir KostićVladimir Kostić
University of Novi SadFaculty of Science
Dept. of Mathematics and Informatics
IntroductionIntroduction
Equivalent definitions of S-SDD Equivalent definitions of S-SDD matricesmatrices
Bounds for the determinantsBounds for the determinants
Convergence area of PDAORConvergence area of PDAOR
Subdirect sumsSubdirect sums
Equivalent definitions of Equivalent definitions of S-SDD matricesS-SDD matrices
S S_
L. Cvetkovic, V. Kostic and R. S. VargaL. Cvetkovic, V. Kostic and R. S. Varga, A new Gersgorin-type eigenvalue A new Gersgorin-type eigenvalue inclusion setinclusion set, Electron. Trans. Numer. Anal., 18 (2004), 73–80Electron. Trans. Numer. Anal., 18 (2004), 73–80
for all Si iir A a i S
for all ,
S S S Si j ii i jj jr A r A a r A a r A
i S j S
1 2
0
max minSj
SSjj jiS SSS j Si Sjii i r A
a r Ar AA A
r Aa r A
Equivalent definitions of Equivalent definitions of S-SDD matricesS-SDD matrices
for all Si iir A a i S
for all ,
S S S Si j ii i jj jr A r A a r A a r A
i S j S
is SDD matrix A S
1 2, S SAJ S A A
is SDD matrix A S
is S-SDD matrix A
Equivalent definitions of Equivalent definitions of S-SDD matricesS-SDD matrices
S S_
S S_
x
1
1 2: ,S SAx J S A A AX is an SDD
H
S-SDD
SDD
x
1S 2
S1
x
Equivalent definitions of Equivalent definitions of S-SDD matricesS-SDD matrices
det Ostrowski 1937 ii ii N
A a r A
det Price 1951 ii ii N
A a u A
det Ouder 1951 ii ii N
A a l A
1
1..1 1
1..
det max ,
max Ostrowski 1952
k n
ii i kk ii ik n
i i k
i i
i nii
A a u A a a l A
l A u A
a
Bounds for the determinantsBounds for the determinants
Bounds for the determinantsBounds for the determinants
12
34
0
100
200
300
400
500
600
700
estimates improved estimates determinant
Bounds for the determinantsBounds for the determinants
12
34
0
100
200
300
400
500
600
700
estimates improved estimates determinant
Bounds for the determinantsBounds for the determinants
12
34
0
100
200
300
400
500
600
700
estimates improved estimates determinant
1 2 ...
1 , 1,2,...N
i
n n n n
n i N
A is block H-matrixiff
M is an M-matrix
A is block H-matrixiff
M is an H-matrix
Convergence of PDAORConvergence of PDAOR
1n
2n
3n
Nn
4n
1n 2n 3n Nn4n
1
, 1,2..ii ij i j NA A A
, 1,2..
1 11 ij
ij i j N
ij ii ij
M m
m A A
1 12m 13m 14m 1Nm
21m 1 23m 24m 2Nm
31m 32m 1 34m 3Nm
41m 42m 43m 1 4Nm
1Nm 2Nm 3Nm 4Nm 1
Convergence of PDAORConvergence of PDAOR
N
, 1...U ij i jA U
, 1...L ij i jA L
1 21,2,..., ...N J J J
1 1 , 1 2 ,...,iJ i i i
, 1...D ij i jA D
1diag ,...D nA D D
1 11 1p pD L D L U D D LX A A A A A A X A A B
D L U DA A A A A
Convergence of PDAORConvergence of PDAOR
Let be block SDD matrix.
Then if we chose parameters in the following way:, 1..
, 2ij i j NA A N
1.. 1..
1.. 1..
1..
1.. 1..
1 11 0 1, min 1 min
2 2
2 1 212 1 2min , 1 min
1 2 2
13 1 2min , 0 1
1 2
1 14 1 2min , min
1
i i
i N i Ni i
i ii
i N i Ni i i
i
i Ni i
i i
i N i Ni
r rand or
l l
r lland or
r l l
land or
r l
l rand
r
0.2 il
PDAOR , 1
L
Lj.Cvetkovic, J. ObrovskiLj.Cvetkovic, J. Obrovski, , Some convergence results of PD relaxation methodsSome convergence results of PD relaxation methods, , AMC 107 (2000) 103-112AMC 107 (2000) 103-112
Convergence of PDAORConvergence of PDAOR
11
1 , )
1
m m m m k sk s J k j
m i j
l A A
1 17 7,7 7,1 7,7 7,2
1 17,7 7,5 7,7 7,6
1 17,7 7,9 7,7 7,10
l A A A A
A A A A
A A A A
Convergence of PDAORConvergence of PDAOR
11
1 , )
1
1
Sm S m m m k s
k s J k j
k s
m i j
l A A
1 14 4,4 4,1 4,4 4,2
1 14,4 4,3 4,4 4,5
Sl A A A A
A A A A
1 18 8,8 8,6 8,8 8,7
1 18,8 8,9 8,8 8,10
18,8 8,11
Sl A A A A
A A A A
A A
Convergence of PDAORConvergence of PDAOR
Let be block S-SDD matrix for set S.
Then if we chose parameters in the following way:
where
, 1.., 2ij i j N
A A N
1 0 1 min , 1 min , ,
2 1 min , 1 min , ,
3 1 min , 0 1,
4 1 min , min , 0.
S SS S
S SS S
S S
S SS S
and or
and or
and or
and
PDAOR , 1
L
, ,
, ,
1 1 1 11min , 2min ,
2 1 1 2 1 2
1 1 1 12min , 2min
11 2 1 2
S S S S S S S Si j i j i j i j
S SS S S S S S S S S Si j S i j Si j i j i i j i i j
S S S S S S S Si j i j i j i j
S SS S S S S Si j S i j Si i j i i j
r r r r l r l r
l r l r r l r r l r
l r l r l r l r
rr l r r l r
,
,1
2 1 2 1 21min .
2 1
S S S Si j i j
S S S S S Si i j i i j
S S S S Si j Si j i j
r r r
r l r r l r
l r l r
CvetkovićCvetković, Kostic, Kostic,, New subclasses of block H-matrices with applications toNew subclasses of block H-matrices with applications toparallel decomposition-type relaxation methodsparallel decomposition-type relaxation methods, , NumeNumer.r. Alg Alg. 42 (2006). 42 (2006)
Convergence of PDAORConvergence of PDAOR
Subdirect sumsSubdirect sums
1 1n nA
2 2n nB
kC A B
Subdirect sumsSubdirect sums
1 2 1 2
11 12
21 22 11 12
21 22
0
0n n k n n k
A A
C A A B B
B B
1 1
11 12
21 22 n n
A AA
A A
22 11, k kA B
2 2
11 12
21 22 n n
B BB
B B
Same sign pattern on
the diagonal
H
SDD
Subdirect sumsSubdirect sums
NO
YESYES
A and B are ,
is the matrix C too?
H
S-SDD
SDD
Subdirect sumsSubdirect sums
R. Bru, F. Pedroche, and D. B. SzyldR. Bru, F. Pedroche, and D. B. Szyld, Subdirect sums of S-Strictly Diagonally Subdirect sums of S-Strictly Diagonally Dominant matricesDominant matrices, Electron. J. Linear Algebra, 15 (2006), 201–209Electron. J. Linear Algebra, 15 (2006), 201–209
NO
YESYES
A and B are ,
is the matrix C too?
Subdirect sumsSubdirect sums
S
Subdirect sumsSubdirect sums
for all ,
S Sii i jj j
S Si j
a r A a r A
r A r A
i S j S
Let
If A is S-SDD and B is SDD then
is S-SDD.
11,2,... ,S card S n k
kC A B
Subdirect sumsSubdirect sums
R. Bru, F. Pedroche, and D. B. SzyldR. Bru, F. Pedroche, and D. B. Szyld, Subdirect sums of S-Strictly Diagonally Subdirect sums of S-Strictly Diagonally Dominant matricesDominant matrices, Electron. J. Linear Algebra, 15 (2006), 201–209Electron. J. Linear Algebra, 15 (2006), 201–209
S
Subdirect sumsSubdirect sums
Subdirect sums of Subdirect sums of SS-SDD matrices-SDD matrices
AS
Subdirect sums of Subdirect sums of SS-SDD matrices-SDD matrices
BS
Subdirect sums of Subdirect sums of SS-SDD matrices-SDD matrices
Let S be arbitrary
If A is - SDD, B is - SDD
and then
is S-SDD.
AS
kC A B
BS
A A B BJ S J S
Thank you for your attentionThank you for your attention