Global root bracketing method with adaptive mesh refinement

TL;DR

This paper introduces an adaptive mesh refinement-based global root bracketing method that efficiently finds all real roots of univariate functions, including multiple roots, with fewer function evaluations.

Contribution

It presents a novel adaptive mesh refinement approach that accurately locates roots, handles multiple roots, and reduces function evaluations compared to existing methods.

Findings

Reduces function evaluations significantly.

Successfully detects roots with even multiplicity.

Demonstrates reliability through multiple examples.

Abstract

An efficient method for finding all real roots of a univariate function in a given bounded domain is formulated. The proposed method uses adaptive mesh refinement to locate bracketing intervals based on bisection criterion for root finding. Each bracketing interval encloses one root. An adaptive form of error is introduced to enclose roots in a desired tolerance based on how much close the roots are. Detecting roots with even multiplicity, which is regarded out of the realm of bracketing methods, becomes possible with the method proposed in this paper. Also, strategies for finding odd-multiple roots with the least number of function evaluations is proposed. Adaptive mesh refinement lead to considerable reduction in function evaluations in comparison to previous global root bracketing methods. The reliability of the proposed method is being illustrated by several examples.