An optimal derivative-free King's family for multiple zeros and its dynamics

Purpose The purpose of this article is to develop and analyze a new derivative-free class of higher-order iterative methods for locating multiple roots numerically. Design/methodology/approach The scheme is generated by using King-type iterative methods. By employing the Traub-Steffensen technique, the proposed class is designed into the derivative-free family. Findings The proposed class requires three functional evaluations at each stage of computation to attain fourth-order convergency. Moreover, it can be observed that the theoretical convergency results of family are symmetrical for particular cases of multiplicity of zeros. This further motivates the authors to present the result in general, which confirms the convergency order of the methods. It is also worth mentioning that the authors can obtain already existing methods as particular cases of the family for some suitable choice of free disposable parameters. Finally, the authors include a wide variety of benchmark problems like van der Waals's equation, Planck's radiation law and clustered root problem. The numerical comparisons are included with several existing algorithms to confirm the applicability and effectiveness of the proposed methods. Originality/value The numerical results demonstrate that the proposed scheme performs better than the existing methods in terms of CPU timing and absolute residual errors.