A new time-efficient and convergent nonlinear solver

Abstract Nonlinear equations arise in various fields of science and engineering. The present era of computational science – where one needs maximum achievement in minimum time – demands proposal of new and efficient iterative methods for solving nonlinear equations and systems. While the new methods are expected to be higher order convergent, the time efficiency and lesser computational information used are the top priorities. In this paper, we propose a new three-step iterative nonlinear solver for nonlinear equations and systems. The proposed method requires three evaluations of function and two evaluations of the first-order derivative per iteration. The proposed method is sixth order convergent, which is also proved theoretically. The performance of the proposed method is tested against other existing methods on the basis of error distributions, computational efficiency and CPU times. The numerical results on the application of the discussed methods on various nonlinear equations and systems, including an application problem related to combustion for a temperature of 3000 °C, show that the proposed method is comparable with existing methods with the main feature of the proposed method being its time-effectiveness.

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