Gergorintype theorems for generalized eigenvalues and their approximations

Geršgorin-type theorems for generalized eigenvalues and their approximations Vladimir Kostić Joint work with Ljiljana Cvetković Richard S. Varga Departman za matematiku i informatiku Univerzitet u Novom Sadu

Short overview. . . o Geršgorin set for generalized eigenvalues o. . . and it’s approximations § § § Stewart’s approximation Cartesian ovals Circles

Short overview. . . o 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 t h eorem. . . Geršgorin 1931

Nonsingularity of matrices. . . SDD Levy 1881 Deplanques 1887 Minkowski 1900 Hadamard 1903

Relationship between these E quivalence ! two statemnts. . . Varga 2004 SDD

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 t h eorem for GEV. . . A is SDD YES/NO YES NO B is SDD YES NO NO

Approximations. . . Stewart 1975 C N O A V R A T L D E S D S S S E s i L C B I R I C A KCV 2010…

Ger š gorin-type ? !

Geršgorin -type ? ! A is GSDD AX is SDD H-MATRICES

Geršgorin -type ? ! H SDD Geršgorin-type localization set

O B S R C R T A A U R U L A O E IL W R N D SI G K O V T LI A E E L C M O S H N CI N A O I S L F Q C I U A Z C E T A E T S SI OI N Geršgorin - t y p e ? ! H alfa_2 alfa_1 SDD DZ Brualdi CKV Generalized Brualdi Varga, R. S. , Lj. , Cvetković, Lj. , Kostić, V. , Approximation of the minimal H-matrix. V. , theory vs. eigenvalue localization. Numerical Cvetković, Kostić, Bru, R. , Pedroche A simple generalization ABetween new eigenvalue localization theorem via Gersgorin and minimal Gersgorin Varga, R. S. , A new. F. , Geršgorin-type eigenvalue Geršgorin set of a square complex matrix, ETNA 30 (2008), 398 -405. Algorithms 42, ETNA 3 -4 (2006), 229 -245. of Gersgorin’s theorem, Advances in Computational Mathematics graph PAMM 5(2005), 787 -788. sets. J. theory, Comput. Appl. Math. 196/2 (2006), 452 -458. inclusion set. (Electronic Transactions on Numerical Analysis) 18 (2009), in print (2004), 73 -80.

DD-type & SDD-type classes. . . o 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 o 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. . . o nonempty class K of square matrices o the set of complex numbers defined as

Isolation principle. . . class K of nonsingular matrices § DD-type class § positively homogenous, i. e. ,

Boundedness principle. . . class K of nonsingular matrices § SDD-type class § positively homogenous, i. e. , YES/NO YES NO NO

Some examples of Geršgorin-type theorems.

Brauer’s Ovals of Cassini Ostrowski 1937 doubly SDD matrices Brauer 1947

BOC for GEV…

Brualdi’s lemniscate sets Brualdi 19 82

Brualdi’s lemniscate sets Brualdi 19 82 ! a ? of air h p p ra rix G at m

Graph of a matrix pair. . .

Brualdi’s lemniscate sets

S-SDD matrices & diag. _sc. S D D S S

S-SDD matrices & diag. sc. S _ S

CKV localization sets for GEV

e G n g š r i r o V K C a r B r e u al i im or n i g m erš G

e G n g š r i r o V K C a r B r e u al i im or n i g m erš G

OPTIMIZATION OF THE POWER CONSUMPTION interference link j 3 1 4 2 Gij link i 7 G= 6 5 9 8 10 D D S 10 x 10 …CKV, H? Power consumption optimization problem has a solution and convergent algorithm that computes the power distribution vector can be obtained 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