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Gergorin-type theorems for generalized eigenvalues and their approximations Departman za matematiku i informatiku Univerzitet u Novom Sadu Vladimir Kosti Joint work with Ljiljana Cvetkovi Richard S. Varga Slide 2 Short overview... Gergorin Gergorin set for generalized eigenvalues...... and and its approximations Stewarts Stewarts approximation Cartesian Cartesian ovals Circles Circles Slide 3 Short overview... Gergorin Gergorin type theorems Definition Definition of the term G-T Th. DD-type DD-type and SDD-type classes of matrices Equivalence Equivalence principle Isolation Isolation principle Boundedness Boundedness principle Some Some of the particular casses Doubly SDD, Brualdi, CKV Slide 4 Slide 5 Gergorins theorem... Gergorin 1931 Slide 6 SDD Levy 1881 Deplanques1887 Deplanques 1887 Minkowski 1900 Hadamard1903 Hadamard 1903 Nonsingularity of matrices... Slide 7 Relationship between these two statemnts... Varga 2004 Equivalence! Slide 8 R. Stewart, Gersgorin theory for generalized eigenvalue problem, Math. Comput. 29 (1975), 600 - 606 Cvetkovi, Lj., Kosti, V., Varga, R.S Gergorin-type localizations of generalized eigenvalues, NLAA (Numerical Linear Algebra with Applications ) 16 (2009), 883 - 898. Slide 9 Gergorins theorem for GEV... Slide 10 Approximations... Stewart 1975 KCV 2010 CIRCLES B is SDD Slide 11 Slide 12 Gergorin-type ?! Slide 13 A is GSDD AX is SDD H- MATRICES Slide 14 H SDD Gergorin-type ?! Gergorin-type localization set Slide 15 H Gergorin-type ?! alfa_1 alfa_2 DZ CKV Brualdi SDD Generalized Brualdi Cvetkovi, Lj., Kosti, V., Varga, R.S., A new Gergorin-type eigenvalue inclusion set. ETNA (Electronic Transactions on Numerical Analysis) 18 (2004), 73-80. Cvetkovi, Lj., Kosti, V., A new eigenvalue localization theorem via graph theory, PAMM 5(2005), 787-788. Cvetkovi, Lj., H-matrix theory vs. eigenvalue localization. Numerical Algorithms 42, 3-4 (2006), 229-245. Cvetkovi, Lj., Kosti, V., Between Gersgorin and minimal Gersgorin sets. J. Comput. Appl. Math. 196/2 (2006), 452-458. Cvetkovi, Lj., Kosti, V., Bru, R., Pedroche F., A simple generalization of Gersgorins theorem, Advances in Computational Mathematics (2009), in print Varga, R.S., Cvetkovi, Lj., Kosti, V., Approximation of the minimal Gergorin set of a square complex matrix, ETNA 30 (2008), 398-405. DDD Slide 16 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 K K K Slide 17 Equivalence principle... nonempty class K of square matrices the set of complex numbers defined as Slide 18 Isolation principle... class K of nonsingular matrices DD-type class positively homogenous, i.e., Slide 19 Boundedness principle... class K of nonsingular matrices SDD-type class positively homogenous, i.e., Slide 20 Slide 21 Brauers Ovals of Cassini Brauer 1947 Ostrowski 1937 doubly SDD matrices Slide 22 BOC for GEV Slide 23 Brualdis lemniscate sets Brualdi 1982 Slide 24 Brualdis lemniscate sets Brualdi 1982 Graph of a matrix pair ?! Slide 25 Graph of a matrix pair... Slide 26 Brualdis lemniscate sets Slide 27 S-SDD matrices & diag. sc. S S _SDD Slide 28 S S _ S S _ Slide 29 CKV localization sets for GEV Slide 30 Gergorin CKV Brauer minimal Slide 31 Brauer CKV minimal Slide 32 link i link j OPTIMIZATION OF THE POWER CONSUMPTION 2 1 3 4 7 6 5 8 9 10 G ij G = 10 x 10 interference 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. Slide 33
Voki Kostic kao izuzetna licnost naseg kulturnog zivota, takodje je jedan od najprefinjenijih gastronoma i sladokusaca. Za ljubitelje dobre hrane sacinio je kuvar koji nam otvara vrata prema raju dobre kuhinje, preporucujuci nam najpikantnija jela - bez mesa.