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Iterative methods of order four and five for solving non-linear system : A study on local convergence
  • Jinny John,
  • Jayakumar Jayaraman
Jinny John
Puducherry Technological University

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Jayakumar Jayaraman
Puducherry Technological University
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Abstract

In this paper, we develop the local convergence analysis of Newton-like fourth and fifth order iterative methods for solving a system of non-linear equations. Earlier studies as in Petkovic (Multipoint methods for solving nonlinear equations, Elsevier, Amsterdam,2013), Traub (Iterative methods for the solution of equations, AMS Chelsea Publishing, Providence, 1982) and Kalyanasundaram et al (International Journal of Applied and Computational Mathematics 3 3:2213-2230, 2017) shows that the local convergence was proved using Taylor series expansion which involved the computation of derivatives of order higher than one. For the fourth and fifth order iterative methods under consideration in this paper, it is required that the functions should be at least five and six times differentiable respectively so that the method is applicable to find the solution. This restricts the applicability of the method and also the cost in finding the solution increases as it involves the computation of higher order derivatives. The local convergence analysis derived in this paper uses Lipschitz and ω-continuity conditions which involves only first derivative (present in the method) to prove the convergence. Moreover, the present study also provides details about the radii of domain of convergence and also estimates on error bounds. Therefore, it is evident that the present study enhances the applicability of the methods under consideration. The obtained results have been verified with suitable numerical illustrations.