# Graph centers used for stabilization of matrix factorizations

• Volume: 30, Issue: 2, page 249-259
• ISSN: 2083-5892

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## Abstract

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Systems of consistent linear equations with symmetric positive semidefinite matrices arise naturally while solving many scientific and engineering problems. In case of a "floating" static structure, the boundary conditions are not sufficient to prevent its rigid body motions. Traditional solvers based on Cholesky decomposition can be adapted to these systems by recognition of zero rows or columns and also by setting up a well conditioned regular submatrix of the problem that is used for implementation of a generalised inverse. Conditioning such a submatrix seems to be related with detection of so called fixing nodes such that the related boundary conditions make the structure as stiff as possible. We can consider the matrix of the problem as an unweighted non-oriented graph. Now we search for nodes that stabilize the solution well-fixing nodes (such nodes are sufficiently far away from each other and are not placed near any straight line). The set of such nodes corresponds to one type of graph center.

## How to cite

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Pavla Kabelíková. "Graph centers used for stabilization of matrix factorizations." Discussiones Mathematicae Graph Theory 30.2 (2010): 249-259. <http://eudml.org/doc/271029>.

@article{PavlaKabelíková2010,
abstract = { Systems of consistent linear equations with symmetric positive semidefinite matrices arise naturally while solving many scientific and engineering problems. In case of a "floating" static structure, the boundary conditions are not sufficient to prevent its rigid body motions. Traditional solvers based on Cholesky decomposition can be adapted to these systems by recognition of zero rows or columns and also by setting up a well conditioned regular submatrix of the problem that is used for implementation of a generalised inverse. Conditioning such a submatrix seems to be related with detection of so called fixing nodes such that the related boundary conditions make the structure as stiff as possible. We can consider the matrix of the problem as an unweighted non-oriented graph. Now we search for nodes that stabilize the solution well-fixing nodes (such nodes are sufficiently far away from each other and are not placed near any straight line). The set of such nodes corresponds to one type of graph center. },
author = {Pavla Kabelíková},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {FETI; parallel computing; generalised inverse; graph center},
language = {eng},
number = {2},
pages = {249-259},
title = {Graph centers used for stabilization of matrix factorizations},
url = {http://eudml.org/doc/271029},
volume = {30},
year = {2010},
}

TY - JOUR
AU - Pavla Kabelíková
TI - Graph centers used for stabilization of matrix factorizations
JO - Discussiones Mathematicae Graph Theory
PY - 2010
VL - 30
IS - 2
SP - 249
EP - 259
AB - Systems of consistent linear equations with symmetric positive semidefinite matrices arise naturally while solving many scientific and engineering problems. In case of a "floating" static structure, the boundary conditions are not sufficient to prevent its rigid body motions. Traditional solvers based on Cholesky decomposition can be adapted to these systems by recognition of zero rows or columns and also by setting up a well conditioned regular submatrix of the problem that is used for implementation of a generalised inverse. Conditioning such a submatrix seems to be related with detection of so called fixing nodes such that the related boundary conditions make the structure as stiff as possible. We can consider the matrix of the problem as an unweighted non-oriented graph. Now we search for nodes that stabilize the solution well-fixing nodes (such nodes are sufficiently far away from each other and are not placed near any straight line). The set of such nodes corresponds to one type of graph center.
LA - eng
KW - FETI; parallel computing; generalised inverse; graph center
UR - http://eudml.org/doc/271029
ER -

## References

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