A Unified Approach to Computing the Zeros of Classical Orthogonal Polynomials

TL;DR

This paper introduces a unified electrostatic-based method for accurately computing zeros of classical orthogonal polynomials, providing error estimates and exact errors for specific polynomial families.

Contribution

It presents a novel, unified approach connecting electrostatic interpretation with energy minimization to compute polynomial zeros.

Findings

Effective method for Jacobi, Laguerre, and Hermite polynomials

Exact error calculations for Chebyshev polynomial zeros

Improved accuracy in zero computation

Abstract

The authors present a unified method for calculating the zeros of the classical orthogonal polynomials based upon the electrostatic interpretation and its connection to the energy minimization problem. Examples are given with error estimates for three cases of the Jacobi polynomials, three cases of the Laguerre polynomials, and the Hermite polynomials. In the case of the Chebyshev polynomials, exact errors are given.