H-matrix theory and its applications

- Ljiljana Cvetkovic
- University of Novi Sad

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

H-matrices

?

?

?

?

H-matrix

M-matrix

Diagonal scaling

A is H-matrix

structure of X

unknown

known

AX is SDD matrix

X

A

Subclasses of H-matrices

aiigt ri

aii(akk- rkaki) gt riaki

Subclasses of H-matrices

aii(akk- rkaki) gt riaki

aiigt ri

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 xgt0

B all nonsingular diagonal matrices

Benefits from H-subclasses

Improving convergence area of relaxation methods

- AOR method
- SDD case convergence area O(A)
- H-case convergence area O(AX)

... next Vladimir Kostic S-SDD Class of

Matrices and its Applications

Here X depends on one real parameter x, which

belongs to an admissible area, so O(AX) T(x)

x1 always included

IMPROVEMENT

Benefits from H-subclasses

Improving bounds for determinants

- Lower bounds
- SDD case det(A) e(A)
- H-case det(A) det(X) e(AX)

... next Vladimir Kostic S-SDD Class of

Matrices and its Applications

e(AX) f(x)

x1 always included

IMPROVEMENT

Benefits from H-subclasses

Simplification of proving matrix properties

Subdirect sums

Schur complement invariants

next after next Maja Kovacevic

Dashnic-Zusmanovich Class of Matrices and its

Applications

Reverse question

Scaling with diagonal matrices of a special form

?

Characterization of new H-subclasses

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

Recent references

Cvetkovic, Kostic, Varga A new Geršgorin type

eigenvalue inclusion area. ETNA 2004

Cvetkovic, Kostic Between Geršgorin and minimal

Geršgorin sets. J. Comput. Appl. Math. 2006

Cvetkovic H matrix Theory vs. Eigenvalue

Localization. Numer. Algor. 2006

Cvetkovic, Kostic New subclasses of block

H-matrices with applications to parallel

decomposition-type relaxation methods. Numer.

Algor. 2006

Cvetkovic, Kostic A note on the convergence of

the AOR method. Appl. Math. Comput. 2007

Future references

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

Algebra in honor of Ivo Marek April 28-30,

2008 Novi Sad

ALA 2005

Thank you!

Dekuji!