NAOct 21, 2014
Solving nonlinear equations by a derivative-free form of the King's family with memorySomayeh Sharifi, Stefan Siegmund, Mehdi Salimi
In this paper, we present an iterative three-point method with memory based on the family of King's methods to solve nonlinear equations. This proposed method has eighth order convergence and costs only four function evaluations per iteration which supports the Kung-Traub conjecture on the optimal order of convergence. An acceleration of the convergence speed is achieved by an appropriate variation of a free parameter in each step. This self accelerator parameter is estimated using Newton's interpolation polynomial of fourth degree. The order of convergence is increased from 8 to 12 without any extra function evaluation. Consequently, this method, possesses a high computational efficiency. Finally, a numerical comparison of the proposed method with related methods shows its effectiveness and performance in high precision computations.
NAOct 19, 2014
Optimal Newton-Secant like methods without memory for solving nonlinear equations with its dynamicsMehdi Salimi, Taher Lotfi, Somayeh Sharifi et al.
We construct two optimal Newton-Secant like iterative methods for solving non-linear equations. The proposed classes have convergence order four and eight and cost only three and four function evaluations per iteration, respectively. These methods support the Kung and Traub conjecture and possess a high computational efficiency. The new methods are illustrated by numerical experiments and a comparison with some existing optimal methods. We conclude with an investigation of the basins of attraction of the solutions in the complex plane.
NAOct 9, 2014
A new class of optimal four-point methods with convergence order 16 for solving nonlinear equationsSomayeh Sharifi, Mehdi Salimi, Stefan Siegmund et al.
We introduce a new class of optimal iterative methods without memory for approximating a simple root of a given nonlinear equation. The proposed class uses four function evaluations and one first derivative evaluation per iteration and it is therefore optimal in the sense of Kung and Traub's conjecture. We present the construction, convergence analysis and numerical implementations, as well as comparisons of accuracy and basins of attraction between our method and existing optimal methods for several test problems.