Geršgorin-type theorems for generalized eigenvalues and their approximations

Departman za matematiku i informatiku

Univerzitet u Novom Sadu

Vladimir Kostić Joint work with

Ljiljana CvetkovićRichard S. Varga

Short overview...

Geršgorin set for generalized eigenvalues

... and it’s approximations Stewart’s approximation Cartesian ovals Circles

Short overview...

Geršgorin type theorems Definition of the term G-T Th. DD-type and SDD-type classes of

matrices Equivalence principle Isolation principle Boundedness principle Some of the particular casses

Doubly SDD, Brualdi, CKV…

GERŠGORIN’S THEOREM...

Geršgorin’s theorem...

G e r š g o r i n 1 9 3 1

Ari

iia

Ai A

ij

iji aAr

,

: , 1, 2,..., .

ii N

i ii i

A A

A z C z a r A i n

SDD

ii ia r A

L e v y 1 8 8 1 D e p l a n q u e s 1 8 8 7 M i n k o w s k i 1 9 0 0 H a d a m a r d 1 9 0 3

Nonsingularity of matrices...

Relationship between these two statemnts...

Ari

iia

Ai A

ij

iji aAr

,

: , 1, 2,..., .

ii N

i ii i

A A

A z C z a r A i n

SDD

Varga 2004

Equivalence!

GERŠGORIN’S THEOREM FOR GENERALIZED EIGENVALUES...

R. Stewart, Gersgorin theory for generalized eigenvalue problem, Math. Comput. 29 (1975), 600 - 606

Cvetković, Lj., Kostić, V., Varga, R.S Geršgorin-type localizations of generalized eigenvalues, NLAA (Numerical Linear Algebra with Applications ) 16 (2009), 883 - 898.

Geršgorin’s theorem for GEV...

A is SDD

B is SDD

YES/NO

YES NO NO

YES NO

ij

jijiiiiii azbazbzBA ,,,, :, C

Ni

i BABA

,,

Approximations...Stewart 1975

i A,B z C : zbi,i ai,i

z21

ri(A) 2 ri(B) 2

KCV 2010…

i A,B z C : zbi,i ai,i z ri(B) ri(A)

z C : zbi,i ai,i z ri(B) ri(B A)

z C : zbi,i ai,i ri(DA DB

1B A)

bi,i ri(B)

CARTESIANOVALS

CIRCLES

B is

SDD

GERŠGORIN-TYPE THEOREMS FOR GENERALIZED EIGENVALUES...

Geršgorin-type ?!

BA,

BA,

0det AzB

SDDnotAzB

BA,KK AzB .

mrnonsingula

of classK

Geršgorin-type ?!

A is GSDD

AX is SDD H - M A T R I C E S

H

BA,K

SDD

Geršgorin-type ?!

BA,ΘK

BA,R

Geršgorin-type local izat ion set

H

Geršgorin-type ?!

a l fa _1

a l fa _2

DZ

CKVBrua ldi

SDD

Genera l i z ed Brua ld i

Cvetković, Lj., Kostić, V., Varga, R.S., A new Geršgorin-type eigenvalue inclusion set. ETNA (Electronic Transactions on Numerical Analysis) 18 (2004), 73-80.

Cvetković, Lj., Kostić, V., A new eigenvalue localization theorem via graph theory, PAMM 5(2005), 787-788.Cvetković, Lj., H-matrix theory vs. eigenvalue localization. Numerical Algorithms 42, 3-4 (2006), 229-245.Cvetković, Lj., Kostić, V.,  Between Gersgorin and minimal Gersgorin sets. J. Comput. Appl. Math. 196/2 (2006), 452-458.Cvetković, Lj., Kostić, V., Bru, R., Pedroche F., A simple generalization of Gersgorin’s theorem, Advances in Computational Mathematics (2009), in print

Varga, R.S., Cvetković, Lj., Kostić, V., Approximation of the minimal Geršgorin set of a square complex matrix, ETNA 30 (2008), 398-405.

OSTROWSKI LOCALIZATIONS

BRAUER OVALS OF CASSINI

BRUALDI LEMN ISCAT ES

DDD

SCAL ING TECHN IQUE

DD-type & SDD-type classes...

K is DD-type class A in K have nonzero diagonal entries A in K iff |A| in K A in K and A B implies B in K

K is SDD-type class K is DD-type class K is opened class, i.e.,

for every A in K, there exists >0,

so that all -perturbations of A remain in the class K

Equivalence principle...

nonempty class K of square matrices

the set of complex numbers defined as

K CK zB-AzBA :,

rnonsingulaare

inmatrices

K

BABA

BAallfor n,n

,,

, K

C

Isolation principle...class K of nonsingular matrices

DD-type class positively homogenous, i.e.,

VUBA

,K

0 , KK AA

ii

ii

b

a

,

,

C

0, iib

Boundedness principle...

class K of nonsingular matrices SDD-type class positively homogenous, i.e.,

YES/NO

YES NO NO

YES NO

BA,K

KAKB

Some examples of Geršgorin-type theorems...

Brauer’s Ovals of Cassini

Brauer 1947

,,

i ji j Ni j

A K A

K

Ostrowski 1937

doubly SDD matr i ces

ArAraa jijjii

, , ,:i j i i j j i jK A z z a z a r A r A C

BOC for GEV…

K i, j A,B z C : zbi,i ai,i zb j, j a j, j

zbi,k ai,k

ki

zb j,k a j,k

kj

K A,B K i, j A,B jN \{i}U

iNU

ˆ K i, j A,B z C : zbi,i ai,i zb j, j a j, j z ri(B) ri(A) z r j (B) r j (A)

Brualdi’s lemniscate sets

Brualdi 19821

2 3

4 567

643216644332211 rrrrraaaaa

545544 rraa

077 a616611 rraa

654321665544332211 rrrrrraaaaaa

Brualdi’s lemniscate sets

Brualdi 1982

i

ii

iiγ razC:z:AB

Acycles

γ AA

BB :

Graph o

f a

matr

ix

pair ?

!

Graph of a matrix pair...

zB A zbi, j ai, j i, jN

A,B ai, j

bi, j

C : bi, j 0, i , j N , i j

C \ A,B

G A,B 1 2 3

4 567

G A G B

Brualdi’s lemniscate sets

i ikkiki

iiiii azbazbC:z:BA ,,,,,B

Acycles

BABA

,:, BB

iii

iiiii ArBrzazbC:z:BA )()(,ˆ

,,B

S-SDD matrices & diag. sc.

ai, i riS A a j, j r j

S A riS A r j

S A

ai, i riS A

i S

S S_

i S, j S

SDD

riS A ai, j

jN \{i}

S-SDD matrices & diag. sc.

S S_

S S_

x

1

SnX

1 2: ,Ax J S A A AX is an SDD

CKV localization sets for GEV

zbi, i ai, i riS zB A zb j, j a j, j r j

S zB A riS zB A r j

S zB A

zbi, i ai, i riS zB A

i S

i S, j S

iS A,B

Vi, jS A,B

C A,B

C S A,B

-10 -5 0 5-4

-2

0

2

4

-10 -5 0 5-4

-2

0

2

4

-10 -5 0 5-4

-2

0

2

4

-10 -5 0 5-4

-2

0

2

4

Geršgor

in

CKV

Brauer

min

imal

Geršgor

in

0 1 2 3 4-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Brauer

0 1 2 3 4-2

-1.5

-1

-0.5

0

0.5

1

1.5

20 1 2 3 4

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

0 1 2 3 4-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

CKV

Geršgor

in

min

imal

Geršgor

in

link i

link j

OPTIMIZATION OF THE POWER CONSUMPTION

2

1

3

4

7

65

8

9

10

Gij

G =

10 x 10

interference

Gi,i Gi, j

ji

Power consumption optimization problem has a solution and convergent algorithm that computes the power distribution vector can be obtained

SDD…CKV, H?

J. Yuan, Z. Li, W. Yu and B. Li, A cross-layer optimization framework for multihop multicast in wireless mesh networks, Journal on Selected Areas in Communications, 24 (2006), 2092-2103.

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