Numerical Solution of Nonlinear Equations pp 117-127 | Cite as

# A numerically stable update for simplicial algorithms

Conference paper

First Online:

## Abstract

In simplicial algorithms, a linear system of equations is solved at each step. Similar to the pivoting steps of linear programming, this can be done by numerically stable techniques [11]. In the following short note, we point out that an even stabler method may be used by looking at the underdetermined linear system involved. The computational cost is less expensive than might be expected since, by the special structure of the labeling, some calculations can be avoided.

## Keywords

Simplicial Algorithm Short Note Stabler Method Stable Technique Penrose Inverse
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## Preview

Unable to display preview. Download preview PDF.

## References

- [1]ALLGOWER, E. and GEORG, K.: Simplicial and continuation methods for approximating fixed points and solutions to systems of equations. SIAM Review 22 (1980) 28–85.MathSciNetCrossRefzbMATHGoogle Scholar
- [2]BEN-ISRAEL,A. and GREVILLE,T.N.E.: Generalized inverses: theory and applications. Wiley-Interscience publ. (1974).Google Scholar
- [3]EAVES, B.C.: A short course in solving equations with PL homotopies. SIAM-AMS Proceedings 9 (1976) 73–143.MathSciNetzbMATHGoogle Scholar
- [4]EAVES, B.C. and SCARF, H.: The solution of systems of piecewise linear equations. Mathematics of Operations Research 1 (1976) 1–27.MathSciNetCrossRefzbMATHGoogle Scholar
- [5]GEORG, K.: Algoritmi simpliciali come realizzazione numerica del grado di Brouwer. In: A survey on the theoretical and numerical trends in nonlinear analysis,I. Gius.Laterza e Figli, Bari (1979) 69–120.Google Scholar
- [6]GILL, P.E., GOLUB, G.H., MURRAY, W. and SAUNDERS, M.A.: Methods for modifying matrix factorizations. Mathematics of Computation 28 (1974) 505–535.MathSciNetCrossRefzbMATHGoogle Scholar
- [7]SCARF, H.E. with HANSEN, T.: Computation of economics equilibria. Yale Univ. Press, New Haven (1973).Google Scholar
- [8]SHERMAN, J. and MORRISON, W.J.: Adjustment of an inverse matrix corresponding to changes in the elements of a given column or a given row of the original matrix. Ann. Math. Statist. 20 (1949) p.621.Google Scholar
- [9]SPANIER,E.H.: Algebraic topology. McGraw-Hill (1966).Google Scholar
- [10]TODD,M.J.: The computation of fixed points and applications. Lecture Notes in Economics and Mathematical Systems 124, Springer-Verlag (1976).Google Scholar
- [11]TODD,M.J.: Numerical stability and sparsity in piecewise linear algorithms. To appear in the proceedings of a symposium on analysis and computation of fixed points, S.M.Robinson (ed.), Academic Press.Google Scholar

## Copyright information

© Springer-Verlag 1981