Abstract
We studied the local convergence of a family of sixth order Jarratt-like methods in Banach space setting. The procedure so applied provides the radius of convergence and bounds on errors under the conditions based on the first Fréchet-derivative only. Such estimates are not proposed in the approaches using Taylor expansions of higher order derivatives which may be nonexistent or costly to compute. In this sense we can extend usage of the methods considered, since the methods can be applied to a wider class of functions. Numerical testing on examples show that the present results can be applied to the cases where earlier results are not applicable. Finally, the convergence domains are assessed by means of a geometrical approach; namely, the basins of attraction that allow us to find members of family with stable convergence behavior and with unstable behavior.
Highlights
We provide local criteria for finding a unique solution δ of the nonlinear equation
For Banach space valued mappings with H : D ⊆ X → Y, and H is differentiable according to Fréchet [1,2]
Alzahrani et al [23] have recently proposed a class of sixth order methods for approximating solution of H ( x ) = 0 using a Jarratt-like composite scheme
Summary
We provide local criteria for finding a unique solution δ of the nonlinear equation. for Banach space valued mappings with H : D ⊆ X → Y, and H is differentiable according to Fréchet [1,2]. Many authors have studied local and semilocal convergence criteria of iterative methods (see, for example [3,4,5,6,7,8,9,10,11,12,13,14]). Alzahrani et al [23] have recently proposed a class of sixth order methods for approximating solution of H ( x ) = 0 using a Jarratt-like composite scheme These methods are very attractive and their local convergence analysis is worthy of study.
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