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

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Short overview... Geršgorin Geršgorin set for generalized eigenvalues...... and and its approximations Stewarts Stewarts approximation Cartesian Cartesian ovals Circles Circles

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Short overview... Geršgorin Geršgorin 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…

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Geršgorins theorem... Geršgorin 1931

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SDD Levy 1881 Deplanques1887 Deplanques 1887 Minkowski 1900 Hadamard1903 Hadamard 1903 Nonsingularity of matrices...

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Relationship between these two statemnts... Varga 2004 Equivalence!

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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.

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Geršgorins theorem for GEV...

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Approximations... Stewart 1975 KCV 2010… CIRCLES B is SDD

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Geršgorin-type ?!

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A is GSDD AX is SDD H- MATRICES

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

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H Geršgorin-type ?! alfa_1 alfa_2 DZ CKV Brualdi SDD Generalized Brualdi 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 Gersgorins 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. DDD

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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

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Equivalence principle... nonempty class K of square matrices the set of complex numbers defined as

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

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

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Brauers Ovals of Cassini Brauer 1947 Ostrowski 1937 doubly SDD matrices

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BOC for GEV…

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Brualdis lemniscate sets Brualdi 1982

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Brualdis lemniscate sets Brualdi 1982 Graph of a matrix pair ?!

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Graph of a matrix pair...

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Brualdis lemniscate sets

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

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S S _ S S _

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CKV localization sets for GEV

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Geršgorin CKV Brauer minimal

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Brauer CKV minimal

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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.

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