Abstract
In this study, we propose an extension of the modified Newton-Househölder methods to find multiple roots with unknown multiplicity of nonlinear equations. With four functional evaluations per iteration, the proposed method achieves an optimal eighth order of convergence. The higher the convergence order, the quicker we get to the root with a high accuracy. The numerical examples have shown that this scheme can compete with the existing methods. This scheme is also stable across all of the functions tested based on the graphical basins of attraction.
Highlights
IntroductionSome modified forms of Househölder’s methods are given in [6,7]
Unknown Multiplicity of NonlinearOne of the most popular problems in mathematics has been finding roots of nonlinear equations f ( x ) = 0
The optimal method we propose for finding multiple roots of unknown multiplicity is eighth-order with four functional evaluations at each iteration
Summary
Some modified forms of Househölder’s methods are given in [6,7] It is hard for such methods to reach an optimal convergence order based on the Kung-Traub hypothesis [8], which states that an iterative scheme can reach the optimal convergence 2k when the number of functional and the derivative evaluations is k + 1. There have been quite a number of methods proposed for finding a multiple root of known multiplicity of nonlinear equations [11,12,13,14,15] Both the root and its multiplicty are unknown. A new fifth-order modified Newton’s method for finding multiple roots of nonlinear equations with unknown multiplicity was developed by Li et al [19]. The optimal method we propose for finding multiple roots of unknown multiplicity is eighth-order with four functional evaluations at each iteration
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