A new efficient and optimal sixteenth-order scheme for simple roots of nonlinear equations

Abstract

[EN] In this manuscript, we propose a new highly efficient and optimal scheme of order sixteen for obtaining simple roots of nonlinear equations. The derivation of this scheme is based on the rational approximation approach. The proposed scheme requires four evaluations of the involved function and one evaluation of its first-order derivative, being optimally consistent with the conjecture of Kung-Traub. In addition, we fully investigated theoretical and computational properties of the proposed scheme along with a main theorem describing the order of convergence. Moreover, we find from the numerical experiments that our proposed methods perform better than the existing optimal sixteenth-order methods when we checked the performance in multi precision digits, on a variety of nonlinear equations.