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H-matrix theory and its applications

H-matrix theory and its applications. Ljiljana Cvetkovi ć University of Novi Sad. Introduction. Subclasses of H-matrices Diagonal scaling Approximation of Minimal Ger š gorin set Improving convergence area of relaxation methods Improving bounds for determinants

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H-matrix theory and its applications

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  1. H-matrix theory and its applications Ljiljana Cvetković University of Novi Sad

  2. Introduction • Subclasses of H-matrices • Diagonal scaling • Approximation of Minimal Geršgorin set • Improving convergence area of relaxation methods • Improving bounds for determinants • Simplification of proving matrix properties • Subdirect sums • Schur complement invariants • Reverse question

  3. ♪ ♪ ♪ -|♪| -|♪| -|♪|  ♪  ♪ ♪ -|♪| -|♪| -|♪| ♪ ♪  ♪ -|♪| -|♪| -|♪| ♪ ♪ ♪  -|♪| -|♪| -|♪| H-matrices || || || || H-matrix M-matrix

  4. Diagonal scaling A is H-matrix structure of X unknown known AX is SDD matrix X A

  5. SDD _ S S Dashnic S-SDD - |aii|> riS |akk|> rkS (|aii|- riS)(|akk|- rkS) > riS rkS - - Subclasses of H-matrices |aii|> ri |aii|(|akk|- rk+|aki|) > ri|aki|

  6. SDD _ S S 1 Dashnic x 1 1 x 1 x 1 x x 1 1 1 S-SDD 1 1 1 1 1 - 1 |aii|> riS |akk|> rkS (|aii|- riS)(|akk|- rkS) > riS rkS 1 1 1 1 - - 1 Subclasses of H-matrices |aii|(|akk|- rk+|aki|) > ri|aki| |aii|> ri

  7. H S-SDD Dash SDD MGS Benefits from H-subclasses Approximation of Minimal Geršgorin set B …explicit forms… B B all diagonal el. 1 except one B all diagonal el. 1 or x>0 B all nonsingular diagonal matrices

  8. SDD case ~ convergence area max Θ(x) x Benefits from H-subclasses Improving convergence area of relaxation methods • AOR method • SDD case ~ convergence area Ω(A) • H-case ~ convergence area Ω(AX) ... next Vladimir Kostić S-SDD Class of Matrices and its Applications   HereXdepends on one real parameterx, which belongs to an admissible area, so Ω(AX) = Θ(x) x=1 always included IMPROVEMENT

  9. k SDD case ~ det(A) ≥ max [max f(x) / xk] x Benefits from H-subclasses Improving bounds for determinants • Lower bounds • SDD case ~ det(A) ≥ ε(A) • H-case ~ det(A) det(X) ≥ ε(AX) ... next Vladimir Kostić S-SDD Class of Matrices and its Applications   ε(AX) = f(x) x=1 always included IMPROVEMENT

  10. Benefits from H-subclasses • Simplification of proving matrix properties • Subdirect sums • Schur complement invariants …next after next Maja Kovačević Dashnic-Zusmanovich Class of Matrices and its Applications

  11. Reverse question • Scaling with diagonal matrices of a special form ? • Characterization of new H-subclasses

  12. Reverse question : YES • Then: • Even better approximation of Minimal Geršgorin set • Furthet improvement of relaxation methods convergence area • Further improvement of bounds for determinants • Simplification of proving more matrix properties

  13. Recent references Cvetković, Kostić, Varga:A new Geršgorin type eigenvalue inclusion area. ETNA2004 Cvetković, Kostić:Between Geršgorinand minimal Geršgorin sets. J. Comput. Appl. Math.2006 Cvetković: Hmatrix Theory vs. Eigenvalue Localization.Numer. Algor.2006 Cvetković, Kostić:New subclasses of block H-matrices with applications to parallel decomposition-type relaxation methods.Numer. Algor.2006 Cvetković, Kostić:A note on the convergence of the AORmethod. Appl. Math. Comput. 2007

  14. Future references… www.im.ns.ac.yu/events/ala2008 Applied Linear Algebra –in honor of Ivo Marek – April 28-30, 2008 Novi Sad

  15. ALA 2005

  16. Thank you! Děkuji!

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