Three-step iterative methods with optimal eighth-order convergence

In this paper, based on Ostrowski's method, a new family of eighth-order methods for solving nonlinear equations is derived. In terms of computational cost, each iteration of these methods requires three evaluations of the function and one evaluation of its first derivative, so that their efficiency indices are 1.682, which is optimal according to Kung and Traub's conjecture. Numerical comparisons are made to show the performance of the new family. © 2011 Elsevier B.V. All rights reserved.This research was supported by Ministerio de Ciencia y Tecnologia MTM2010-18539.Cordero Barbero, A.; Torregrosa Sánchez, JR.; Penkova Vassileva, M. (2011). Three-step iterative methods with optimal eighth-order convergence. Journal of Computational and Applied Mathematics. 235(10):3189-3194. https://doi.org/10.1016/j.cam.2011.01.004S318931942351

Solving Polynomial Equations using Modified Super Ostrowski Homotopy Continuation Method

Homotopy continuation methods (HCMs) are now widely used to find the roots of polynomial equations as well as transcendental equations.  HCM can be used to solve the divergence problem as well as starting value problem. Obviously, the divergence problem of traditional methods occurs when a method cannot be operated at the beginning of iteration for some points, known as bad initial guesses. Meanwhile, the starting value problem occurs when the initial guess is far away from the exact solutions.   The starting value problem has been solved using Super Ostrowski homotopy continuation method for the initial guesses between . Nevertheless, Super Ostrowski homotopy continuation method was only used to find out real roots of nonlinear equations.  In this paper, we employ the Modified Super Ostrowski-HCM to solve several real life applications which involves polynomial equations by expanding the range of starting values. The results indicate that the Modified Super Ostrowski-HCM performs better than the standard Super Ostrowski-HCM. In other words, the complex roots of polynomial equations can be found even the starting value is real with this proposed scheme

Minimization of Nonlinear Functions by Certain Numerical Algorithms

Abstract: In this paper, we propose few new algorithms, for minimization of nonlinear functions. Then comparative study among the new algorithms and Newton's algorithm is established by means of various examples

On Constructing Two-Point Optimal Fourth-Order Multiple-Root Finders with a Generic Error Corrector and Illustrating Their Dynamics

With an error corrector via principal branch of the mth root of a function-to-function ratio, we propose optimal quartic-order multiple-root finders for nonlinear equations. The relevant optimal order satisfies Kung-Traub conjecture made in 1974. Numerical experiments performed for various test equations demonstrate convergence behavior agreeing with theory and the basins of attractions for several examples are presented