A modified Chebyshev’s iterative method with at least sixth order of convergence

Local convergence for a derivative free method of order three under weak conditions

“…which shows equation (23) Using equation (15) we get that ( ) 3 The radius r A was shown by us to be the convergence radius of Newton's method (Amat et al, 2008;Argyros, 2008;Argyros and Hilout, 2010) ( ) ( ) under the conditions (15) and (16). It follows from the definition of r that the convergence radius r of the method (2) cannot be larger than the convergence radius r A of the second order Newton's method.…”

“…That is why many authors have used higher order multi-point methods (Ahmad et al, 2009;Amat et al, 2008;Argyros, 2008;Argyros and Hilout, 2010;Bruns and Bailey, 1977;Marquina, 1990a, 1990b;Chun, 1990;Ezquerro and Hernández, 2000, 2005Gutiérrez and Hernández, 1998;Ganesh and Joshi, 1991;Hernández, 2001;Hernández and Salanova, 1999;Kantorovich and Akilov, 1982;Gupta, 2007, 2010;Parida and Gupta, 2007;Ren et al, 2009;Rheinboldt, 1977;Traub, 1964;Wang et al, 2009Wang et al, , 2011Ye and Li, 2006;Ye et al, 2007;Zhao and Wu, 2008;Kou, 2012a, 2012b;Zhu and Wu, 2003). In this paper, we present the local convergence of the derivative free method defined for each n = 0, 1, 2, … by…”

“…Third order methods such as Euler's, Halley's, super Halley's, Chebyshev's (Ahmad et al, 2009;Amat et al, 2008;Argyros, 2008;Argyros and Hilout, 2010;Bruns and Bailey, 1977;Marquina, 1990a, 1990b;Chun, 1990;Ezquerro and Hernández, 2000, 2005Gutiérrez and Hernández, 1998;Ganesh and Joshi, 1991;Hernández, 2001;Hernández and Salanova, 1999;Kantorovich and Akilov, 1982;Gupta, 2007, 2010;Parida and Gupta, 2007;Ren et al, 2009;Rheinboldt, 1977;Traub, 1964;Wang et al, 2009Wang et al, , 2011Ye and Li, 2006;Ye et al, 2007;Zhao and Wu, 2008;Kou, 2012a, 2012b;Zhu and Wu, 2003) require the evaluation of the second derivative F″ at each step, which in general is very expensive. That is why many authors have used higher order multi-point methods (Ahmad et al, 2009;Amat et al, 2008;Argyros, 2008;Argyros and Hilout, 2010;Bruns and Bailey, 1977;Marquina, 1990a, 1990b;Chun, 1990;Ezquerro and Hernández, 2000, 2005Gutiérrez and Hernández, 1998;Ganesh and Joshi, 1991;Hernández, 2001;Hernández and Salanova, 1999;Kantorovich and Akilov...…”

Local Convergence for Some Third-Order Iterative Methods Under Weak Conditions

“…(5) It is worth noticing that the method (1.4) is not changing when we use the conditions of Theorem 2.1 instead of the stronger conditions used in [1], [9]- [25], [27]- [30]. Moreover, we can compute the computational order of convergence (COC) defined by ξ = ln |x n+1 − x * | |x n − x * | ln |x n − x * | |x n−1 − x * | or the approximate computational order of convergence ξ 1 = ln |x n+1 − x n | |x n − x n−1 | ln |x n − x n−1 | |x n−1 − x n−2 | .…”

Efficient high-order methods based on golden ratio for nonlinear systems

“…During the last years, numerous papers devoted to iterative methods for solving nonlinear systems have appeared in several journals. Some methods existing in the literature are based on the use of interpolation quadrature formulas (see [1][2][3][4]), or include the second partial derivative of the function F or different estimations of it (see [5][6][7][8]), or are Steffensen's type methods (see [9]), etc. We also pay attention to the known Jarratt's method (J) (see [10]) whose efficiency is widely recognized.…”