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G eršgorin -type theorems for generalized eigenvalues and their approximations

G eršgorin -type theorems for generalized eigenvalues and their approximations. Vladimir Kostić. Joint work with Ljiljana Cvetkovi ć Richard S. Varga. Departm a n za matematiku i informatiku Univer zitet u Nov om Sad u. Short overview . Geršgorin set for generalized eigenvalues

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G eršgorin -type theorems for generalized eigenvalues and their approximations

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  1. Geršgorin-type theorems for generalized eigenvalues and their approximations VladimirKostić Joint work with LjiljanaCvetković Richard S. Varga Departman za matematiku i informatiku Univerzitetu Novom Sadu

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

  3. 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…

  4. Geršgorin’s theorem...

  5. Geršgorin’s theorem... Geršgorin 1931

  6. SDD Nonsingularity of matrices... Levy 1881 Deplanques1887 Minkowski1900 Hadamard1903

  7. Relationship between these two statemnts... Equivalence! Varga2004 SDD

  8. Geršgorin’s theoremfor 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.

  9. Geršgorin’s theorem for GEV... YES NO YES/NO NO NO YES

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

  11. Geršgorin-type theorems for generalized Eigenvalues...

  12. Geršgorin-type ?!

  13. Geršgorin-type ?! A isGSDD AX is SDD H-MATRICES

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

  15. OSTROWSKI LOCALIZATIONS Geršgorin-type ?! BRAUER OVALSOF CASSINI SCALING TECHNIQUE BRUALDI LEMNISCATES alfa_2 H alfa_1 SDD DDD DZ Brualdi CKV Generalized Brualdi Varga, R.S., Cvetković, Lj., Kostić, V.,Approximation of the minimal Geršgorin set of a square complex matrix, ETNA 30 (2008), 398-405. Cvetković, Lj.,H-matrix theory vs. eigenvalue localization. Numerical Algorithms 42, 3-4 (2006), 229-245. Cvetković, Lj., Kostić, V., Bru, R., Pedroche F., A simple generalization of Gersgorin’s theorem, Advances in Computational Mathematics (2009), in print Cvetković, Lj., Kostić, V., A new eigenvalue localization theorem via graph theory, PAMM 5(2005), 787-788. Cvetković, Lj., Kostić, V.,Between Gersgorin and minimal Gersgorin sets. J. Comput. Appl. Math. 196/2 (2006), 452-458. Cvetković, Lj., Kostić, V., Varga, R.S., A new Geršgorin-type eigenvalue inclusion set. ETNA (Electronic Transactions on Numerical Analysis) 18 (2004), 73-80.

  16. DD-type & SDD-type classes... • K is DD-type class • Ain Khave nonzero diagonal entries • Ain Kiff|A| in K • Ain Kand A   B implies B in K • K is SDD-type class • K is DD-type class • K is opened class, i.e., for every Ain K, there exists >0, so that all -perturbations of A remain in the class K

  17. Equivalence principle... • nonempty classK of square matrices • the set of complex numbers defined as

  18. Isolation principle... • classK of nonsingular matrices • DD-type class • positively homogenous, i.e.,

  19. Boundedness principle... • classK of nonsingular matrices • SDD-type class • positively homogenous, i.e., YES NO YES/NO NO NO YES

  20. Some examples of Geršgorin-type theorems...

  21. Brauer’s Ovals of Cassini Ostrowski1937 Brauer1947 doubly SDDmatrices

  22. BOC for GEV…

  23. Brualdi’s lemniscate sets Brualdi1982

  24. Brualdi’s lemniscate sets Brualdi1982 Graph of a matrix pair ?!

  25. Graph of a matrix pair...

  26. Brualdi’s lemniscate sets

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

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

  29. CKV localization sets for GEV

  30. Geršgorin CKV Brauer minimal Geršgorin

  31. Geršgorin Brauer CKV minimal Geršgorin

  32. OPTIMIZATION OF THE POWER CONSUMPTION 3 link j interference 1 4 Gij 2 link i 7 G = 6 5 9 SDD 10 x10 …CKV, H? 10 8 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 (2006), 2092-2103.

  33. Thank you for your attention...

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