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Mathematics of Computation

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Computation of topological degree using interval arithmetic, and applications

Author: Oliver Aberth
Journal: Math. Comp. 62 (1994), 171-178
MSC: Primary 65G10; Secondary 55M25
MathSciNet review: 1203731
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Abstract: A method is described for computing the topological degree of a mapping from $ {R^n}$ into $ {R^n}$ 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, $ {B^{(1)}}$, 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 $ {B^{(1)}}$. Again a portion, $ {B^{(2)}}$, of the boundary of $ {B^{(1)}}$ 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 $ {B^{(n - 1)}}$.

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 $ f(x)$ defined over an interval [a, b], using sign changes of $ f(x)$, is easily generalized to apply to n functions defined over boxes, using the topological degree. For this application we actually use the topological degree $ \bmod\;2$, 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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Article copyright: © Copyright 1994 American Mathematical Society

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