Abstract
A general scheme of third order convergence is described for finding multiple roots of nonlinear equations. The proposed scheme requires one evaluation of f, f′, and f′′ each per iteration and contains several known one-point third order methods for finding multiple roots, as particular cases. Numerical examples are included to confirm the theoretical results and demonstrate convergence behavior of the proposed methods. In the end, we provide the basins of attraction for some methods to observe their dynamics in the complex plane.
Highlights
Solving nonlinear equations is one of the most important problems in numerical analysis [1, 2]
We provide the basins of attraction for some methods to observe their dynamics in the complex plane
In order to improve the order of convergence of (1), several third order methods with known multiplicity m have been proposed at the expense of an additional evaluation of second derivative
Summary
Solving nonlinear equations is one of the most important problems in numerical analysis [1, 2]. In order to improve the order of convergence of (1), several third order methods with known multiplicity m have been proposed at the expense of an additional evaluation of second derivative (see [4,5,6,7,8,9,10,11,12,13,14]). Sharma [12, 13] can be regarded as particular cases of the proposed family.
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