Three-Point Iterated Interval Half-Cutting for Finding All Local Minima of Unknown Single-Variable Function

Abstract

A numerical method is suggested to find all local minima and the global minimum of an unknown single-variable function bounded on a given interval regardless of the interval length. The method has six inputs: three inputs defined straightforwardly and three inputs, which are adjustable. The endpoints of the initial interval and a formula for evaluating the single-variable function at any point of this interval are the straightforward inputs. The three adjustable inputs are a tolerance with the minimal and maximal numbers of subintervals. The tolerance is the secondary adjustable input. Having broken the initial interval into a set of subintervals, the three-point iterated half-cutting “gropes” around every local minimum by successively cutting off a half of the subinterval or dividing the subinterval in two. A range of subinterval sets defined by the minimal and maximal numbers of subintervals is covered by running the threepoint half-cutting on every set of subintervals. As a set of values of currently found local minima points changes less than by the tolerance, the set of local minimum points and the respective set of function values at these points are returned. The presented approach is applicable to whichever task of finding local extrema is. If primarily the purpose is to find all local maxima or the global maximum of the function, the presented approach is applied to the function taken with the negative sign. The presented approach is a significant and important contribution to the field of numerical estimation and approximate analysis. Although the method does not assure obtaining all local minima (or maxima) for any function, setting appropriate minimal and maximal numbers of subintervals makes missing some minima (or maxima) very unlikely.