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On the Convergence, Dynamics and Applications of a New Class of Nonlinear System Solvers

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Abstract

In this paper, a uni-parametric class of third-order iterative algorithms for solving systems of nonlinear equations is proposed. The local convergence of the suggested schemes is studied using generalized Lipschitz-type condition on the first-order Fréchet derivative. Furthermore, we analyze the numerical stability of the new methods applying complex dynamics tool. The nonlinear systems related to the equation of molecular interaction, a boundary value problem, the integral equation from Chandrasekhar’s work, etc. are discussed. The most interesting fact about the proposed third-order family is that it generates a super convergent scheme (for \(\gamma =2\)) for solving quadratic nonlinear systems (QNS). This particular method produces much better results for QNS in comparison with other third-order schemes. Also, it is observed that the approximate computational order of convergence (ACOC) of this scheme (for \(\gamma =2\)) is approximately four (3.99–4.00) while solving QNS.

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Acknowledgements

The second author would like to thank University Grants Commission (UGC) of India for the financial support (ID: NOV2017-402662).

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Authors and Affiliations

  1. Department of Mathematical Sciences, Cameron University, Lawton, OK, 73505, USA

    Ioannis K. Argyros

  2. Department of Mathematics, IIIT Bhubaneswar, Bhubaneswar, Odisha, 751003, India

    Debasis Sharma & Shanta Kumari Sunanda

  3. Department of Mathematics, Fakir Mohan University, Balasore, Odisha, 756020, India

    Sanjaya Kumar Parhi

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  1. Ioannis K. Argyros
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  4. Shanta Kumari Sunanda
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Correspondence to Debasis Sharma.

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Argyros, I.K., Sharma, D., Parhi, S.K. et al. On the Convergence, Dynamics and Applications of a New Class of Nonlinear System Solvers. Int. J. Appl. Comput. Math 6, 142 (2020). https://doi.org/10.1007/s40819-020-00893-4

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