A numerically stable update for simplicial algorithms
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 . 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.
KeywordsSimplicial Algorithm Short Note Stabler Method Stable Technique Penrose Inverse
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- BEN-ISRAEL,A. and GREVILLE,T.N.E.: Generalized inverses: theory and applications. Wiley-Interscience publ. (1974).Google Scholar
- 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
- SCARF, H.E. with HANSEN, T.: Computation of economics equilibria. Yale Univ. Press, New Haven (1973).Google Scholar
- 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
- SPANIER,E.H.: Algebraic topology. McGraw-Hill (1966).Google Scholar
- TODD,M.J.: The computation of fixed points and applications. Lecture Notes in Economics and Mathematical Systems 124, Springer-Verlag (1976).Google Scholar
- 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