H-matrix theory H-matrix theory and its applicationsand its applications

Ljiljana CvetkoviLjiljana Cvetkovićć

University of Novi SadUniversity of Novi Sad

IntroductionIntroduction

Subclasses of H-matrices Subclasses of H-matrices Diagonal scalingDiagonal scaling

Approximation of Minimal GerApproximation of Minimal Gerššgorin setgorin set Improving convergence area of relaxation Improving convergence area of relaxation

methodsmethods Improving bounds for determinantsImproving bounds for determinantsSimplification of proving matrix propertiesSimplification of proving matrix properties

Subdirect sumsSubdirect sums Schur complement invariantsSchur complement invariants

Reverse questionReverse question

H-matricesH-matrices

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H-matrixH-matrix M-matrixM-matrix

Diagonal scalingDiagonal scaling

A is H-matrixA is H-matrix

AX is SDD matrixAX is SDD matrix

AA XX

structure of Xstructure of X

unknownunknown knownknown

Subclasses of H-matricesSubclasses of H-matricesS S

_

SDD

|aii|(|akk|- rk+|aki|) > ri|aki|

Dashnic

S-SDD

|aii|> ri

|aii|> riS |akk|> rk

S

(|aii|- riS)(|akk|- rk

S) > riS rk

S -

-

-

Subclasses of H-matricesSubclasses of H-matricesS S

_

SDD

|aii|(|akk|- rk+|aki|) > ri|aki|

Dashnic

S-SDD

|aii|> ri

x

x

1

1

1

1

1

xx

1

1

1

1

1

1

1

x1

1

1

1

1

1

1

|aii|> riS |akk|> rk

S

(|aii|- riS)(|akk|- rk

S) > riS rk

S -

-

-

Benefits from H-Benefits from H-subclassessubclasses

Approximation of Minimal GerApproximation of Minimal Gerššgorin gorin setset

( )A

( )D A

( )C A

MGS

H

B all nonsingular

diagonal matrices

B all diagonal

el. 1 or x>0

B all diagonal el. 1

except one

B

B

S-SDD

Dash

SDD

…explicit forms…

Benefits from H-Benefits from H-subclassessubclasses

Improving convergence area of relaxation Improving convergence area of relaxation methodsmethodsAOR method AOR method

SDD case ~ convergence area SDD case ~ convergence area ΩΩ(A)(A)

H-case ~ convergence area H-case ~ convergence area ΩΩ(A(AXX))

HereHere XX depends on one real parameterdepends on one real parameter xx, , which belongs to an which belongs to an admissible admissible area, so area, so ΩΩ(A(AXX) = ) = ΘΘ((xx)) SDD case ~ convergence area max SDD case ~ convergence area max ΘΘ((xx))

xx

x=1 always x=1 always includedincluded

IMPROVEMENTIMPROVEMENT

... next... next

Vladimir KostiVladimir Kostić ć   

S-SDD Class of Matrices and its S-SDD Class of Matrices and its Applications  Applications  

Benefits from H-Benefits from H-subclassessubclasses

Improving bounds for Improving bounds for determinantsdeterminants

Lower boundsLower bounds

SDD case ~ SDD case ~ det(A)det(A) ≥ ≥ εε(A)(A)

H-case ~ H-case ~ det(A) det(A) det(det(XX)) ≥ ≥ εε(A(AXX))

εε(A(AXX) = f() = f(xx))

x=1 always x=1 always includedincluded

IMPROVEMENTIMPROVEMENT

... next... next

Vladimir KostiVladimir Kostić ć   

S-SDD Class of Matrices and its S-SDD Class of Matrices and its Applications  Applications  

SDD case ~ det(A) SDD case ~ det(A) ≥≥ max [max f( max [max f(xx) / ) / xxkk]] xxkk

Benefits from H-Benefits from H-subclassessubclassesSimplification of proving matrix Simplification of proving matrix propertiesproperties

Subdirect sumsSubdirect sums

Schur complement invariantsSchur complement invariants

……next after nextnext after next

Maja Kovačević Maja Kovačević   

Dashnic-Zusmanovich Class of Matrices Dashnic-Zusmanovich Class of Matrices and its Applicationsand its Applications

Reverse questionReverse question

Scaling with diagonal matrices of a Scaling with diagonal matrices of a special form special form

Characterization of new H-subclassesCharacterization of new H-subclasses

??

Reverse question : YESReverse question : YES

Then: Then:

Even better approximation of Minimal GerEven better approximation of Minimal Gerššgorin gorin setset Furthet improvement of relaxation methods Furthet improvement of relaxation methods convergence area convergence area Further improvement of bounds for determinantsFurther improvement of bounds for determinants Simplification of proving more matrix propertiesSimplification of proving more matrix properties

Recent referencesRecent references

Cvetković, Kostić: Between Geršgorin and minimal Geršgorin sets. J. Comput. Appl. Math. 2006

Cvetković, Kostić, Varga: A new Geršgorin type eigenvalue inclusion area. ETNA 2004

Cvetković, Kostić: A note on the convergence of the AOR method. Appl. Math. Comput. 2007

Cvetković, Kostić: New subclasses of block H-matrices with applications to parallel decomposition-type relaxation methods. Numer. Algor. 2006

Cvetković: H matrix Theory vs. Eigenvalue Localization. Numer. Algor. 2006

www.im.ns.ac.yu/events/ala2008

Applied Linear AlgebraApplied Linear Algebra

–– in honor of Ivo Marek – in honor of Ivo Marek –

April 28-30, 2008 Novi SadApril 28-30, 2008 Novi Sad

Future references…Future references…

ALA 2005ALA 2005

Thank you!Thank you!

DěkujiDěkuji!!