A study on the local convergence and the dynamics of Chebyshev–Halley–type methods free from second derivative

Abstract

We study the local convergence of Chebyshev-Halley-type methods of convergence order at least five to approximate a locally unique solution of a nonlinear equation. Earlier studies such as Behl (2013), Bruns and Bailey (Chem. Eng. Sci 32, 257–264, 1977), Candela and Marquina (Computing 44, 169–184, 1990), (Computing 45(4):355–367, 1990), Chicharro et al. (2013), Chun (Appl. Math. Comput, 190(2):1432–1437, 1990), Cordero et al. (Appl.Math. Lett. 26, 842–848, 2013), Cordero et al. (Appl. Math. Comput. 219, 8568–8583, 2013), Cordero and Torregrosa (Appl. Math. Comput. 190, 686–698, 2007), Ezquerro and Hernández (Appl. Math. Optim. 41(2):227–236, 2000), (BIT Numer. Math. 49, 325–342, 2009), (J. Math. Anal. Appl. 303, 591–601, 2005), Gutiérrez and Hernández (Comput. Math. Applic. 36(7):1–8, 1998), Ganesh and Joshi (IMA J. Numer. Anal. 11, 21–31, 1991), Hernández (Comput. Math. Applic. 41(3–4):433–455, 2001), Hernández and Salanova (Southwest J. Pure Appl. Math. 1, 29–40, 1999), Jarratt (Math. Comput. 20(95):434–437, 1996), Kou and Li (Appl. Math. Comput. 189, 1816–1821, 2007), Li (Appl. Math. Comput. 235, 221–225, 2014), Ren et al. (Numer. Algorithm. 52(4):585–603, 2009), Wang et al. (Numer. Algorithm. 57, 441–456, 2011), Kou et al. (Numer. Algorithm. 60, 369–390, 2012) show convergence under hypotheses on the third derivative or even higher. The convergence in this study is shown under hypotheses on the first derivative. Hence, the applicability of the method is expanded. The dynamical analyses of these methods are also studied. Finally, numerical examples are also provided to show that our results apply to solve equations in cases where earlier studies cannot apply.