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An Efficient Class of Weighted-Newton Multiple Root Solvers with Seventh Order Convergence

Department of Mathematics, Sant Longowal Institute of Engineering & Technology, Longowal, Sangrur 148106, India
Engineering School (DEIM), University of Tuscia, 01100 Viterbo, Italy
Ton Duc Thang University, Ho Chi Minh City (HCMC) 758307, Vietnam
Author to whom correspondence should be addressed.
Symmetry 2019, 11(8), 1054; (registering DOI)
Received: 27 June 2019 / Revised: 7 August 2019 / Accepted: 13 August 2019 / Published: 16 August 2019
(This article belongs to the Special Issue Symmetry and Complexity 2019)
PDF [937 KB, uploaded 16 August 2019]


In this work, we construct a family of seventh order iterative methods for finding multiple roots of a nonlinear function. The scheme consists of three steps, of which the first is Newton’s step and last two are the weighted-Newton steps. Hence, the name of the scheme is ‘weighted-Newton methods’. Theoretical results are studied exhaustively along with the main theorem describing convergence analysis. Stability and convergence domain of the proposed class are also demonstrated by means of using a graphical technique, namely, basins of attraction. Boundaries of these basins are fractal like shapes through which basins are symmetric. Efficacy is demonstrated through numerical experimentation on variety of different functions that illustrates good convergence behavior. Moreover, the theoretical result concerning computational efficiency is verified by computing the elapsed CPU time. The overall comparison of numerical results including accuracy and CPU-time shows that the new methods are strong competitors for the existing methods. View Full-Text
Keywords: nonlinear equations; multiple roots; higher order methods; attraction basins nonlinear equations; multiple roots; higher order methods; attraction basins

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Sharma, J.R.; Kumar, D.; Cattani, C. An Efficient Class of Weighted-Newton Multiple Root Solvers with Seventh Order Convergence. Symmetry 2019, 11, 1054.

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