Chebyshev's approximation algorithms and applications

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

“…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...…”

Semilocal Convergence Analysis of an Iteration of Order Five Using Recurrence Relations in Banach Spaces

“…Various modification of Newton's method are proposed to increase the order of convergence and efficiency. In literature [11,18,13,2,15,8,9,16], authors have established the semilocal convergence of higher order iterative methods under various continuity conditions. Recently, the semilocal convergence of an efficient fifth order method is established in [17] under Lipschitz condition on F .…”

Local convergence of deformed Halley method in Banach space under Holder continuity conditions

“…Iterative methods of convergence order higher than two such as Chebyshev-Halley-type methods [1,3,5,7]- [16] require the evaluation of the second Fréchet-derivative, which is very expensive in general. However, there are integral equations, where the second Fréchet-derivative is diagonal by blocks and inexpensive [10]- [13] or for quadratic equations the second Fréchet-derivative is constant [4,12]. Moreover, in some applications involving stiff systems [2], [5], [9], high order methods are usefull.…”