Mehdi Salimi

Gunar Matthies, Mehdi Salimi, Somayeh Sharifi et al.

We present a three-point iterative method without memory for solving nonlinear equations in one variable. The proposed method provides convergence order eight with four function evaluations per iteration. Hence, it possesses a very high computational efficiency and supports Kung and Traub's conjecture. The construction, the convergence analysis, and the numerical implementation of the method will be presented. Using several test problems, the proposed method will be compared with existing methods of convergence order eight concerning accuracy and basin of attraction. Furthermore, some measures are used to judge methods with respect to their performance in finding the basin of attraction.

Gunar Matthies, Mehdi Salimi, Somayeh Sharifi et al.

In this paper, we present a three-point without memory iterative method based on Kung and Traub's method for solving non-linear equations in one variable. The proposed method has eighth-order convergence and costs only four function evaluations each iteration which supports the Kung-Traub conjecture on the optimal order of convergence. Consequently, this method possesses very high computational efficiency. We present the construction, the convergence analysis, and the numerical implementation of the method. Furthermore, comparisons with some other existing optimal eighth-order methods concerning accuracy and basins of attraction for several test problems will be given.

Massimiliano Ferrara, Somayeh Sharifi, Mehdi Salimi

In this paper, we modify the Newton-Secant method with third order of convergence for finding multiple roots of nonlinear equations. Per iteration this method requires two evaluations of the function and one evaluation of its first derivative. This method has the efficiency index equal to $3^{\frac{1}{3}}\approx 1.44225$. We describe the analysis of the proposed method along with numerical experiments including comparison with existing methods. Moreover, the dynamics of the proposed method are shown with some comparisons to the other existing methods.

Somayeh Sharifi, Massimiliano Ferrara, Mehdi Salimi et al.

In this paper, we present a family of three-point with eight-order convergence methods for finding the simple roots of nonlinear equations by suitable approximations and weight function based on Maheshwari method. Per iteration this method requires three evaluations of the function and one evaluation of its first derivative. This class of methods has the efficiency index equal to $8^{\frac{1}{4}}\approx 1.682$. We describe the analysis of the proposed methods along with numerical experiments including comparison with existing methods.

Somayeh 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.

Mehdi 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.

Somayeh 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.