# On the existence and approximation of zeroes

- 47 Downloads
- 9 Citations

## Abstract

In this paper we consider the problem of finding zeroes of a continuous function*f* from a convex, compact subset*U* of ℝ^{ n } to ℝ^{ n }. In the first part of the paper it is proved that*f* has a computable zero if*f*:*C* ^{ n }→ℝ^{ n } satisfies the nonparallel condition for any two antipodal points on bd*C* ^{n}, i.e. if for any*x*∈bd*C* ^{ n },*f(x)≠αf(−x)*, α≥0, holds. Therefore we describe a simplicial algorithm to approximate such a zero. It is shown that generally the degree of the approximate zero depends on the number of reflection steps made by the algorithm, i.e. the number of times the algorithm switches from a face τ on bd*C* ^{ n } to the face −τ. Therefore the index of a terminal simplex σ is defined which equals the local Brouwer degree of the function if σ is full-dimensional. In the second part of the paper the algorithm is used to generate possibly several approximate zeroes of*f*. Two sucessive solutions may have both the same or opposite degrees, again depending on the number of reflection steps. By extending*f*:*U*→ℝ^{ n } to a function g from a cube containing*U* to ℝ^{ n }, the procedure can be applied to any continuous function*f* without having any information about the global and local Brouwer degrees a priori.

## Key words

Simplical Approximation Complementary 1-Simplex Variable Dimension Algorithm## Preview

Unable to display preview. Download preview PDF.

## References

- [1]E.I. Allgower and K. Georg, “Simplicial and continuation methods for approximating fixed points and solutions to systems of equations”,
*SIAM Review*22 (1980) 28–75.zbMATHCrossRefMathSciNetGoogle Scholar - [2]E.L. Allgower and K. Georg, “Homotopy methods for approximating several solutions to nonlinear systems of equations”, in: W. Forster, ed.,
*Numerical solution of highly nonlinear problems*(North-Holland, Amsterdam, 1980) pp. 253–270.Google Scholar - [3]U. Borsuk, “Drei Sätze über die
*n*-dimensionale euklidische Sphär”,*Fundamenta Mathematica*20 (1933) 177–190.zbMATHGoogle Scholar - [4]R.M. Freund, “Variable dimension complexes with applications”, Technical Report Sol 80-11, Department of Operations Research, Stanford University, Stanford, CA, 1980).Google Scholar
- [5]R.M. Freund and M.J. Todd, “A constructive proof of Tucker's combinatorial lemma”,
*Journal of Combinatorial Theory Series A*30 (1981), 321–325.zbMATHCrossRefMathSciNetGoogle Scholar - [6]C.B. Garcia and W.I. Zangwill, “Global continuation methods for finding all solutions to polynomial systems of equations in
*n*variables”, in: A.V. Fiacco and K.O. Kortenak, eds.*Extremal methods and systems analysis*, Lecture Notes in Economics & Mathematical Systems 174 (Springer, Berlin, 1980) pp. 481–497.Google Scholar - [7]M.M. Jeppson, “A search for the fixed points of a continuous mapping”, in: R.H. Day and S.M. Robinson, eds.,
*Mathematical topics in economics theory and computation*(SIAM, Philadelphia, 1972) pp. 122–129.Google Scholar - [8]M.A. Krasnosell'skii,
*Topological methods in the theory of nonlinear integral equations*(Pergamon Press, Oxford, 1964).Google Scholar - [9]H.W. Kuhn, “Finding roots by pivoting”, in: S. Karamardian, ed.,
*Fixed Points: algorithms and applications*(Academic Press, New York, 1977) pp. 11–39.Google Scholar - [10]G. van der Laan, with the collaboration of A.J.J. Talman, Simplicial fixed point algorithms (Mathematical Centre Tract 129, Mathematisch Centrum, Amsterdam, 1980).zbMATHGoogle Scholar
- [11]G. van der Laan and A.J.J. Talman, “A restart algorithm for computing fixed points without an extra dimension”,
*Mathematical Programming*17 (1979) 74–84.zbMATHCrossRefMathSciNetGoogle Scholar - [12]G. van der Laan and A.J.J. Talman, “A restart algorithm without an artificial level for computing fixed points on unbounded regions”, in: H.O. Peitgen and H.O. Walther, eds.,
*Functional differential equations and approximation of fixed points*(Lecture Notes in Mathematics 730, Springer, Berlin, 1979) pp. 247–256.CrossRefGoogle Scholar - [13]G. van der Laan and A.J.J. Talman, “A class of simplicial restart fixed point algorithms without an extra dimension”,
*Mathematical Programming*20 (1981) 33–48.zbMATHCrossRefMathSciNetGoogle Scholar - [14]G. van der Laan and A.J.J. Talman, “Labelling rules and orientation: on Sperner's lemma and Brouwer degree”, in: E.L. Allgower, K. Glashoff and H.O. Peitgen, eds.,
*Numerical solution of nonlinear equations*, Lecture Notes in Mathematics 878 (Springer, 1981) pp. 238–257.Google Scholar - [15]S. Lefschetz,
*Introduction to Topology*(Princeton University Press, Princeton, NJ, 1949).zbMATHGoogle Scholar - [16]L.A. Lyusternik and L.G. Shnirel'man, “Topological methods in variational problems and their applications to the differential geometry of surfaces”,
*Uspekhi Matematicheskikh Nauk (N.S.)*2 (1947).Google Scholar - [17]O.H. Merrill, “Applications and extensions of an algorithm that computes fixed points of certain upper semi-continuous point to set mappings”, Ph.D. Dissertation, University of Michigan (Ann Arbor, MI, 1972).Google Scholar
- [18]M. Prüfer and H.W. Siegberg, “On computational aspects of topological degree in
*R*^{n}”, in: H.O. Peitgen and H.O. Walther, eds.,*Functional differential equations and approximations of fixed points*, Lecture Notes in Mathematics 730 (Springer, Berlin 1979) pp. 410–433.CrossRefGoogle Scholar - [19]P.M. Reiser, “A modified integer labelling for complementarity algorithms”,
*Mathematics of Operations Research*6 (1981) 129–139.zbMATHMathSciNetCrossRefGoogle Scholar - [20]A.J.J. Talman, with the collaboration of G. van der Laan, Variable dimension fixed point algorithms and triangulations, Mathematical Centre Tract 128 (Mathematisch Centrum, Amsterdam, 1980).zbMATHGoogle Scholar
- [21]M.J. Todd,
*The computation of fixed points and applications*(Lecture Notes in Economics and Mathematical Systems 124, Springer, Berlin, 1976).zbMATHGoogle Scholar - [22]M.J. Todd, “Global and local convergence and monotonicity results for a recent variable-dimension simplicial algorithm”, in: W. Forster, ed.,
*Numerical solution of highly nonlinear problems*(North-Holland, Amsterdam, 1980) pp. 43–69.Google Scholar - [23]M.J. Todd and A.M. Wright, “A variable dimension simplicial algorithm for antipodal fixed point theorems”,
*Numerical Functional Analysis and Optimization*2 (1980) 155–186.zbMATHMathSciNetGoogle Scholar - [24]A. W. Tucker, “Some topological properties of disk and sphere”,
*Proceedings of the First Canadian Mathematical Congress*(University of Toronto Press, Toronto, 1946) pp. 285–309.Google Scholar