A new modified Halley method without second derivatives for nonlinear equation

On the Higher-Order Method for the Solution of a Nonlinear Scalar Equation

“…Due to this, we can always find a neighborhood of α such that f (1) (x k ) = 0. Thus, we can choose in this neighborhood…”

“…By using Taylor series and (30), we can also write f (z k ) and f (1) (z k ) in terms of α and z k :…”

“…f (α) = 0. In the rest of the paper, it is assumed that f (α) = 0 ( 1 ) and the given function will be smooth. Moreover, it is assumed that the starting point x 0 will be close to α, when discussing the convergence rate of an algorithm.…”

“…In [1], the author presents a modified Halley method which has a quintic convergence by adding a right predictorcorrector in the updating formula.…”

Fifth and Sixth-Order Iterative Algorithms Without Derivatives for Solving Non-Linear Equations

“…(FCAM1) and (FCAM2) are compared with methods that do not use derivatives such as Steffensens (SM), Cordero's (CM), Jain's (JM) [4] and Zheng's methods (ZM) [14]. We also compare methods that use derivatives such as Newton's (NM), Chun's (CHM) [2], Kou's (KM) [5] and Noor's methods (NOM) [7]. All computations were carried out with double arithmetic precision on MATLAB 10.…”