Abstract
We present a local convergence analysis for eighth-order variants of Newton’s method in order to approximate a solution of a nonlinear equation. We use hypotheses up to the first derivative in contrast to earlier studies such as Amat et al. (Appl Math Comput 206(1):164–174, 2008), Amat et al. (Aequationes Math 69:212–213, 2005), Chun et al. (Appl Math Comput. 227:567–592, 2014), Petkovic et al. (Multipoint methods for solving nonlinear equations. Elsevier, Amsterdam, 2013), Potra and Ptak (Nondiscrete induction and iterative processes. Pitman Publ, Boston, 1984), Rall (Computational solution of nonlinear operator equations. Robert E. Krieger, New York, 1979), Ren et al. (Numer Algorithms 52(4):585–603, 2009), Rheinboldt (An adaptive continuation process for solving systems of nonlinear equations. Banach Center, Warsaw, 1975), Traub (Iterative methods for the solution of equations. Prentice Hall, Englewood Cliffs, 1964), Weerakoon and Fernando (Appl Math Lett 13:87–93, 2000), Wang and Kou (J Differ Equ Appl 19(9):1483–1500, 2013) using hypotheses up to the seventh derivative. This way the applicability of these methods is extended under weaker hypotheses. Moreover, the radius of convergence and computable error bounds on the distances involved are also given in this study. Numerical examples are also presented in this study.
Highlights
In this study, we are concerned with the problem of approximating a locally unique solution x∗ of equation F(x) = 0, (1.1)where F : D ⊆ S → S is a nonlinear function, D is a convex subset of S and S is R or C
We present the local convergence analysis of method (1.2)
We presented a new local convergence analysis for an eighth-order method for solving equations based on contractive techniques and Lipschitz constants under hypotheses only on the first derivative
Summary
We are concerned with the problem of approximating a locally unique solution x∗ of equation. We study the local convergence of eighth-order method defined for each n = 0, 1, 2, . We provide a radius of convergence and computable error estimates on the distances using |xn − x∗| with Lipschitz constants not provided in the earlier studies by Taylor expansions This way we expand the applicability of method (1.2). For the local convergence analysis that follows we define some functions and parameters. We can compute the computational order of convergence (COC) defined by ξ = ln This way we obtain in practice the order of convergence in a way that avoids the bounds involving estimates using estimates higher than the first Fréchet derivative of operator F
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