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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

H-matrix theory and its applications

Ljiljana Cvetković

University of Novi Sad

introduction
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-matrices

||

||

||

||

H-matrix

M-matrix

diagonal scaling
Diagonal scaling

A is H-matrix

structure of X

unknown

known

AX is SDD matrix

X

A

subclasses of h matrices

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|

subclasses of h matrices6

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

benefits from h subclasses

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

benefits from h subclasses8

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

benefits from h subclasses9

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

benefits from h subclasses10
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

reverse question
Reverse question
  • Scaling with diagonal matrices of a special form

?

  • Characterization of new H-subclasses
reverse question yes
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
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

future references
Future references…

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

Applied Linear Algebra

–in honor of Ivo Marek –

April 28-30, 2008 Novi Sad

thank you
Thank you!

Děkuji!