Finite and Constructive Conditions for a Solution to f(x) = 0.
CHICAGO UNIV ILL CENTER FOR MATHEMATICAL STUDIES IN BUSINESS AND ECONOMICS
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In R sup 1, if a continuous function has opposite signs at the end-point of an interval, then the function has a zero in the interval. If the function has a nonvanishing derivative at a zero, then there is an interval such that the function has opposite signs at the endpoints. In this paper each of these results is extended to R sup n. The extension relates the existence of a zero of a function to its behavior at a finite number of points in terms of opposite sign conditions analogous to those in R sup 1. The sufficient conditions for the existence of a zero will be necessary when the function has a nonsingular Jacobian matrix at the zero. The sufficiency proof is constructive and provides a computational procedure for finding an approximate solution to fx 0. The results presented have implications for questions relating to the existence and computation of solutions to other problems such as finding fixed points of continuous functions in R sup n, or finding solutions to the complementarity problem.
- Operations Research