A Bisection Method Like Algorithm for Approximating Extrema of a Continuous Function

TL;DR

This paper introduces a bisection-like algorithm for approximating the extrema of continuous functions on closed domains, analyzing its convergence, discussing its advantages and disadvantages, and demonstrating its application with Matlab code.

Contribution

The paper presents a novel bisection-like algorithm for finding function extrema, including convergence analysis and practical implementation details.

Findings

The algorithm converges with a specific order.

Error bounds are established at each iteration.

The method performs well on certain classes of functions.

Abstract

For a continuous function $f$ defined on a closed and bounded domain, there is at least one maximum and one minimum. First, we introduce some preliminaries which are necessary through the paper. We then present an algorithm, which is similar to the bisection method, to approximate those maximum and minimum values. We analyze the order of the convergence of the method and the error at the $k$-th step. Then we discuss the pros and cons of the method. Finally, we apply our method for some special classes of functions to obtain nicer results. At the end, we write a Matlab script which implements our algorithm.