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Constructive Theory of Designing Optimal Eighth-Order Derivative-Free Methods for Solving Nonlinear Equations
Institute of Mathematics and Digital Technology, Mongolian Academy of Sciences, Ulaanbaatar, Mongolia
School of Applied Sciences, Mongolian University of Science and Technology, Ulaanbaatar, Mongolia
Department of Informatics, Mongolian National University of Education, Ulaanbaatar, Mongolia
- 1 Institute of Mathematics and Digital Technology, Mongolian Academy of Sciences, Ulaanbaatar, Mongolia
- 2 School of Applied Sciences, Mongolian University of Science and Technology, Ulaanbaatar, Mongolia
- 3 Department of Informatics, Mongolian National University of Education, Ulaanbaatar, Mongolia
American Journal of Computational Mathematics·Volume 10 (2020)·Pages 100–117·Published 7 January 2020·DOI10.4236/ajcm.2020.101007
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Abstract
This paper stresses the theoretical nature of constructing the optimal derivative-free iterations. We give necessary and sufficient conditions for derivative-free three-point iterations with the eighth-order of convergence. We also establish the connection of derivative-free and derivative presence three-point iterations. The use of the sufficient convergence conditions allows us to design wide class of optimal derivative-free iterations. The proposed family of iterations includes not only existing methods but also new methods with a higher order of convergence.
KeywordsMultipoint MethodsDerivative-Free MethodsOrder of Convergence
- Thukral, R. and Petković, M.S. (2010) A Family of Three-Point Methods of Optimal Order for Solving Nonlinear Equations. The Journal of Computational and Applied Mathematics, 233, 2278-2284. https://doi.org/10.1016/j.cam.2009.10.012
- Rhee, M.S., Kim, Y.I. and Neta, B. (2018) An Optimal Eighth-Order Class of Three-Step Weighted Newton’s Methods and Their Dynamics behind the Purely Imaginary Extraneous Fixed Points. International Journal of Computer Mathematics, 95, 2174-2211. https://doi.org/10.1080/00207160.2017.1367387
- Petković, M.S., Neta, B., Petković, L.D. and Dzunic, J. (2014) Multipoint Methods for Solving Nonlinear Equations. Applied Mathematics and Computation, 226, 635-660. https://doi.org/10.1016/j.amc.2013.10.072
- Argyros, I.K., Kansal, M., Kanwar, V. and Bajaj, S. (2017) Higher-Order Derivative-Free Families of Chebyshev-Halley Type Methods with or without Memory for Solving Nonlinear Equations. Applied Mathematics and Computation, 315, 224-245. https://doi.org/10.1016/j.amc.2017.07.051
- Soleymani, F. and Khattri, S.K. (2012) Finding Simple Roots by Seventh- and Eighth-Order Derivative-Free Methods. International Journal of Mathematical Models and Methods in Applied Sciences, 6, 45-52. https://doi.org/10.1155/2012/932420
- Matthies, G., Salimi, M., Sharifi, S. and Varona, J.L. (2016) An Optimal Three-Point Eighth-Order Iterative Method without Memory for Solving Nonlinear Equations with Its Dynamics. Japan Journal of Industrial and Applied Mathematics, 33, 751-766. https://doi.org/10.1007/s13160-016-0229-5
- Thukral, R. (2011) Eighth-Order Iterative Methods without Derivatives for Solving Nonlinear Equations. International Scholarly Research Network ISRN Applied Mathematics, 2011, Article ID: 693787. https://doi.org/10.5402/2011/693787 https://www.hindawi.com/journals/isrn/2011/693787
- Soleymani, F. and Vanani, S.K. (2011) Optimal Steffensen-Type Methods with Eighth Order of Convergence. Computers & Mathematics with Applications, 62, 4619-4626. https://doi.org/10.1016/j.camwa.2011.10.047
- Zhanlav, T., Ulziibayar, V. and Chuluunbaatar, O. (2017) Necessary and Sufficient Conditions for the Convergence of Two and Three-Point Newton-Type Iterations. Computational Mathematics and Mathematical Physics, 57, 1090-1100. https://doi.org/10.1134/S0965542517070120
- Zhanlav, T., Chuluunbaatar, O. and Ulziibayar, V. (2017) Generating Function Method for Constructing New Iterations. Applied Mathematics and Computation, 315, 414-423. https://doi.org/10.1016/j.amc.2017.07.078