S-SDD class of matrices and its application Vladimir Kostić University of Novi Sad Faculty of Science Dept. of Mathematics and Informatics

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


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


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


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



12

34

0

100

200

300

400

500

600

700

estimates improved estimates determinant


12

34

0

100

200

300

400

500

600

700



12

34

0

100

200

300

400

500

600

700


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


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


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


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


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


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)



1 1n nA

2 2n nB

kC A B


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


NO

YESYES

A and B are ,

is the matrix C too?

H

S-SDD

SDD


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?


S


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


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 sums of Subdirect sums of SS-SDD matrices-SDD matrices

AS


BS


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

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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