A study on the local convergence and complex dynamics of Kou’s family of iterative methods

Abstract

Obtaining convergence domain is an important task in the study of iterative schemes. Analysis of local convergence of an iterative procedure provides essential information about its convergence domain around a solution. In this manuscript, we study the local analysis of the uni-parametric Kou’s class of iterative algorithms for addressing nonlinear equations. This approach expands the utility of the methods by preventing the use of Taylor expansion in convergence analysis. In the view of extending the applicability of these methods, the convergence analysis is shown using Lipschitz condition on the first derivative. Our study provides radii of convergence balls and the uniqueness of the solution along with the calculable error distances. The complex dynamical analysis of the family is also presented. Numerical examples are solved to show that our theoretical conclusions work well in the situation where the earlier analysis cannot be implemented.

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Correspondence to Debasis Sharma.

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Argyros, I.K., Sharma, D., Parhi, S.K. et al. A study on the local convergence and complex dynamics of Kou’s family of iterative methods. SeMA (2021). https://doi.org/10.1007/s40324-021-00257-y

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Keywords

  • Nonlinear equations
  • Iterative methods
  • Local convergence
  • Lipschitz continuity
  • Parameter space
  • Dynamical plane

Mathematics Subject Classification

  • 37F10
  • 65D99
  • 65G99
  • 65J15
  • 65Y20