ISSN 1088-6842(online) ISSN 0025-5718(print)

Computation of topological degree using interval arithmetic, and applications

Author: Oliver Aberth
Journal: Math. Comp. 62 (1994), 171-178
MSC: Primary 65G10; Secondary 55M25
DOI: https://doi.org/10.1090/S0025-5718-1994-1203731-4
MathSciNet review: 1203731
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Abstract: A method is described for computing the topological degree of a mapping from into defined by n functions of n variables on a region specified as a product of n intervals, a generalized box B. The method is an adaptation of Kearfott's method to boxes, and begins by checking the signs of the n functions on the boundary of B with interval arithmetic. On the basis of this check, a portion, , of the boundary of B is designated for further investigation, and one of the n functions defining the mapping is dropped. The signs of the remaining functions are checked on the boundary of . Again a portion, , of the boundary of is designated for further investigation, and another of the functions is dropped. On the nth cycle of the process, the topological degree finally is evaluated by determining the signs of a single function on a collection of isolated points, comprising the boundary of a region .

When the topological degree is nonzero, there is at least one point inside B where the n functions are simultaneously zero. To locate such a point, the familiar bisection method for functions defined over an interval [a, b], using sign changes of , is easily generalized to apply to n functions defined over boxes, using the topological degree. For this application we actually use the topological degree , the crossing parity, because its computation is easier. If the n functions have all partial derivatives in the box B, with a nonzero Jacobian at any point where the functions are simultaneously zero, then all such points inside B can be located by another method, which also uses the crossing parity.

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DOI: https://doi.org/10.1090/S0025-5718-1994-1203731-4
Article copyright: © Copyright 1994 American Mathematical Society