A New Initial Approximation Bound in the Durand Kerner Algorithm for Finding Polynomial Zeros

TL;DR

This paper proposes two new bounds for initial approximations in the Durand-Kerner algorithm, improving convergence stability and speed in polynomial root-finding through theoretical and numerical validation.

Contribution

Introduces the lambda maximal bound and New bound 1 for initial approximations, enhancing the Durand-Kerner algorithm's convergence reliability and efficiency.

Findings

Lambda maximal bound ensures roots lie within a complex circle.

New bound 1 guarantees convergence but with larger radii.

Numerical experiments confirm improved stability and speed.

Abstract

The Durand-Kerner algorithm is a widely used iterative technique for simultaneously finding all the roots of a polynomial. However, its convergence heavily depends on the choice of initial approximations. This paper introduces two novel approaches for determining the initial values: New bound 1 and the lambda maximal bound, aimed at improving the stability and convergence speed of the algorithm. Theoretical analysis and numerical experiments were conducted to evaluate the effectiveness of these bounds. The lambda maximal bound consistently ensures that all the roots lie within the complex circle, leading to faster and more stable convergence. Comparative results demonstrate that while New bound 1 guarantees convergence, but it yields excessively large radii.