Numerical Algorithms

, Volume 77, Issue 4, pp 1249–1272

# An eighth-order family of optimal multiple root finders and its dynamics

Original Paper

## Abstract

There is a very small number of higher-order iteration functions for multiple zeros whose order of convergence is greater than four. Some scholars have tried to propose optimal eighth-order methods for multiple zeros. But, unfortunately, they did not get success in this direction and attained only sixth-order convergence. So, as far as we know, there is not a single optimal eighth-order iteration function in the available literature that will work for multiple zeros. Motivated and inspired by this fact, we present an optimal eighth-order iteration function for multiple zeros. An extensive convergence study is discussed in order to demonstrate the optimal eighth-order convergence of the proposed scheme. In addition, we also demonstrate the applicability of our proposed scheme on real-life problems and illustrate that the proposed methods are more efficient among the available multiple root finding techniques. Finally, dynamical study of the proposed schemes also confirms the theoretical results.

## Keywords

Nonlinear equations Optimal iterative methods Multiple roots Efficiency index Kung-Traub conjecture

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## Authors and Affiliations

• Ramandeep Behl
• 1
• Alicia Cordero
• 2
• Sandile S. Motsa
• 1
• 3
• Juan R. Torregrosa
• 2
1. 1.School of Mathematics, Statistics and Computer SciencesUniversity of KwaZulu-NatalScottsvilleSouth Africa
2. 2.Instituto Universitario de Matemática MultidisciplinarUniversitat Politècnica de ValènciaValènciaSpain
3. 3.Mathematics DepartmentUniversity of SwazilandKwaluseniSwaziland