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Article

Convergence Analysis and Complex Geometry of an Efficient Derivative-Free Iterative Method

by 1,2,*,†, 1,*,† and 3,4,*
1
Department of Mathematics, Sant Longowal Institute of Engineering and Technology, Longowal 148106, Sangrur, India
2
Chandigarh University, Gharuan 140413, Mohali, India
3
Department of Physics and Chemistry, Technical University of Cluj-Napoca, Cluj-Napoca 400114, Romania
4
Institute of Doctoral Studies, Babeş-Bolyai University, Cluj-Napoca 400084, Romania
*
Authors to whom correspondence should be addressed.
These authors contributed equally to this work.
Mathematics 2019, 7(10), 919; https://doi.org/10.3390/math7100919
Received: 12 September 2019 / Revised: 27 September 2019 / Accepted: 29 September 2019 / Published: 2 October 2019
(This article belongs to the Special Issue Numerical Methods)
To locate a locally-unique solution of a nonlinear equation, the local convergence analysis of a derivative-free fifth order method is studied in Banach space. This approach provides radius of convergence and error bounds under the hypotheses based on the first Fréchet-derivative only. Such estimates are not introduced in the earlier procedures employing Taylor’s expansion of higher derivatives that may not exist or may be expensive to compute. The convergence domain of the method is also shown by a visual approach, namely basins of attraction. Theoretical results are endorsed via numerical experiments that show the cases where earlier results cannot be applicable. View Full-Text
Keywords: local convergence; nonlinear equations; Banach space; Fréchet-derivative local convergence; nonlinear equations; Banach space; Fréchet-derivative
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MDPI and ACS Style

Kumar, D.; Sharma, J.R.; Jäntschi, L. Convergence Analysis and Complex Geometry of an Efficient Derivative-Free Iterative Method. Mathematics 2019, 7, 919. https://doi.org/10.3390/math7100919

AMA Style

Kumar D, Sharma JR, Jäntschi L. Convergence Analysis and Complex Geometry of an Efficient Derivative-Free Iterative Method. Mathematics. 2019; 7(10):919. https://doi.org/10.3390/math7100919

Chicago/Turabian Style

Kumar, Deepak; Sharma, Janak R.; Jäntschi, Lorentz. 2019. "Convergence Analysis and Complex Geometry of an Efficient Derivative-Free Iterative Method" Mathematics 7, no. 10: 919. https://doi.org/10.3390/math7100919

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