Geršgorin-type theorems for generalized eigenvalues and their approximations Departman za...
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Transcript of Geršgorin-type theorems for generalized eigenvalues and their approximations Departman za...
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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