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.
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References
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.
BEN-ISRAEL,A. and GREVILLE,T.N.E.: Generalized inverses: theory and applications. Wiley-Interscience publ. (1974).
EAVES, B.C.: A short course in solving equations with PL homotopies. SIAM-AMS Proceedings 9 (1976) 73–143.
EAVES, B.C. and SCARF, H.: The solution of systems of piecewise linear equations. Mathematics of Operations Research 1 (1976) 1–27.
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.
GILL, P.E., GOLUB, G.H., MURRAY, W. and SAUNDERS, M.A.: Methods for modifying matrix factorizations. Mathematics of Computation 28 (1974) 505–535.
SCARF, H.E. with HANSEN, T.: Computation of economics equilibria. Yale Univ. Press, New Haven (1973).
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.
SPANIER,E.H.: Algebraic topology. McGraw-Hill (1966).
TODD,M.J.: The computation of fixed points and applications. Lecture Notes in Economics and Mathematical Systems 124, Springer-Verlag (1976).
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.
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© 1981 Springer-Verlag
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Georg, K. (1981). A numerically stable update for simplicial algorithms. In: Allgower, E.L., Glashoff, K., Peitgen, HO. (eds) Numerical Solution of Nonlinear Equations. Lecture Notes in Mathematics, vol 878. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0090679
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DOI: https://doi.org/10.1007/BFb0090679
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