An Optimal Derivative-Free Ostrowski’s Scheme for Multiple Roots of Nonlinear Equations

Abstract

Finding higher-order optimal derivative-free methods for multiple roots (m≥2) of nonlinear expressions is one of the most fascinating and difficult problems in the area of numerical analysis and Computational mathematics. In this study, we introduce a new fourth order optimal family of Ostrowski’s method without derivatives for multiple roots of nonlinear equations. Initially the convergence analysis is performed for particular values of multiple roots—afterwards it concludes in general form. Moreover, the applicability and comparison demonstrated on three real life problems (e.g., Continuous stirred tank reactor (CSTR), Plank’s radiation and Van der Waals equation of state) and two standard academic examples that contain the clustering of roots and higher-order multiplicity (m=100) problems, with existing methods. Finally, we observe from the computational results that our methods consume the lowest CPU timing as compared to the existing ones. This illustrates the theoretical outcomes to a great extent of this study.

Keywords

• nonlinear equation
• King–Traub conjecture
• multiple root
• optimal iterative method
• efficiency index