Abstract
Gaussian elimination is studied as the elimination process in the bipartite graph associated with the matrix of the system, and the concept of reachable sets applied to bipartite graphs is introduced. A characterization of the elimination graphs in terms of reachable sets is presented for the case of bipartite graphs. A bipartite quotient graph model is given, which allows the computation of reachable sets during the elimination process in a more advantageous way. An implementation of the bipartite quotient graph model as well as the advantages of applying quotient graphs over elimination graphs are considered. This extends to the unsymmetric case the theory of George and Liu (5).
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Bibliography
Bareiss E. and Maulino C. "Finite Sparse Matrix Techniques" DOE EV-76-S-02-2280. (1979).
Duff I.S. "A survey of Sparse Matrix Research", Harwell Report CSS28 (1976).
—. "Practical Comparison of Codes for the Solution of Sparse Linear Systems". Sparse Matrix Proceedings. SIAM (Duff I.S. and Stewart G.W. Eds), (1978).
George A. and Liu J. Computer Solution of Large Sparse Positive Definite Systems. Prentice Hall. Series in Computational Mathematics, (1980).
—. "An Optimal Algorithm for Symbolic Factorization of Symmetric Matrices" Research Report CS78-11. Department of Computer Science, University of Waterloo, Waterloo, Ontario (1978).
Golumbic M.C. "Perfect Elimination and chordal bipartite graphs". Journal of Graph Theory. 2 (1978) 155–163.
Parter S. "The Use of Linear Graphs in Gauss Elimination". SIAM Review 3 (1961) 119–130.
Reid J.S. "Solution of Linear Systems of Equations: Direct Methods (General)". Lectures Notes in Mathematics Sparse Matrix Techniques. (A Dold and Eckman Eds). Springer-Verlag, Berlin, 1976.
Rose D.J. and Tarjan R.E. "Algorithmic Aspects of Vertex Elimination on Directed Graphs". SIAM J. Appl. Math. 34 (1978) 176–197.
Tarjan R. "Depth-First Search and Linear Graph Algorithms". SIAM J. Comput (1972) 146–160.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 1983 Springer-Verlag
About this paper
Cite this paper
Pagallo, G., Maulino, C. (1983). A bipartite quotient graph model for unsymmetric matrices. In: Numerical Methods. Lecture Notes in Mathematics, vol 1005. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0112537
Download citation
DOI: https://doi.org/10.1007/BFb0112537
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-12334-7
Online ISBN: 978-3-540-40967-0
eBook Packages: Springer Book Archive