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Axioms 2019, 8(2), 37; https://doi.org/10.3390/axioms8020037

Efficient Two-Step Fifth-Order and Its Higher-Order Algorithms for Solving Nonlinear Systems with Applications

Department of Mathematics, Pondicherry Engineering College, Pondicherry 605014, India
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Both authors have read and approved the final manuscript.
Received: 20 February 2019 / Revised: 20 March 2019 / Accepted: 26 March 2019 / Published: 1 April 2019
PDF [336 KB, uploaded 1 April 2019]

Abstract

This manuscript presents a new two-step weighted Newton’s algorithm with convergence order five for approximating solutions of system of nonlinear equations. This algorithm needs evaluation of two vector functions and two Frechet derivatives per iteration. Furthermore, it is improved into a general multi-step algorithm with one more vector function evaluation per step, with convergence order 3 k + 5 , k 1 . Error analysis providing order of convergence of the algorithms and their computational efficiency are discussed based on the computational cost. Numerical implementation through some test problems are included, and comparison with well-known equivalent algorithms are presented. To verify the applicability of the proposed algorithms, we have implemented them on 1-D and 2-D Bratu problems. The presented algorithms perform better than many existing algorithms and are equivalent to a few available algorithms.
Keywords: Newton’s method; system of nonlinear equations; higher-order method; multi-step method; Frechet derivative; computational efficiency Newton’s method; system of nonlinear equations; higher-order method; multi-step method; Frechet derivative; computational efficiency
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).
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Sivakumar, P.; Jayaraman, J. Efficient Two-Step Fifth-Order and Its Higher-Order Algorithms for Solving Nonlinear Systems with Applications. Axioms 2019, 8, 37.

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