# On the existence and approximation of zeroes

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## 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

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