H-matrix theory and its applications

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H-matrix theory and its applications

• Ljiljana Cvetkovic
• University of Novi Sad

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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

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H-matrices
?
?
?
?
H-matrix
M-matrix
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Diagonal scaling
A is H-matrix
structure of X
unknown
known
AX is SDD matrix
X
A
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Subclasses of H-matrices
aiigt ri
aii(akk- rkaki) gt riaki
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Subclasses of H-matrices
aii(akk- rkaki) gt riaki
aiigt ri
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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
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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
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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
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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
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Reverse question
Scaling with diagonal matrices of a special form
?
Characterization of new H-subclasses
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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

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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
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Future references
www.im.ns.ac.yu/events/ala2008 Applied Linear
Algebra in honor of Ivo Marek April 28-30,
2008 Novi Sad
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ALA 2005
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Thank you!
Dekuji!