THE EXTERNAL AGGREGATION NEWTON’S METHOD FOR SOLVING NONLINEAR EQUATIONS AND APPLICATIONS

Marija Paunović; Dejan Ćebić; Nebojša Ralević

doi:10.5937/univtho10-24982

Marija Paunović Faculty of Hotel Management and Tourism in Vrnjacka Banja, University of Kragujevac, Vrnjacka Banja, Serbia
Dejan Ćebić Faculty of Mining and Geology, University of Belgrade, Belgrade, Serbia
Nebojša Ralević Faculty of Technical Sciences, University of Novi Sad, Novi Sad, Serbia

DOI: https://doi.org/10.5937/univtho10-24982

Keywords: Newton’s method, Aggregation function, Order of convergence

Abstract

Different modified Newton’s methods are widely used in modern computational engineering science, applied mechanics, economics, optimization problems and other disciplines. On the other hand aggregation of information takes a significant place in many knowledge-based systems, where aggregation of data or values is needed. In this paper we theoretically analysed a new modification of Newton’s method based on aggregation function applied on finding multiple roots of nonlinear equations, and numerically verified theoretical results on the examples with simple and multiple roots. Numerical analysis of the proposed approach with obtained results and the related discussion are presented on examples in practice.

References

Babajee, D. K. R. & Dauhoo, M. Z. 2006, An analysis of the properties of the variants of Newton’s method with third order convergence, Applied Mathematics and Computation, 183(1), pp. 659-684. doi:10.1016/j.amc.2006.05.116

Douglas, J. M. 1972. Process Dynamics and Control: Control System Synthesis, Prentice Hall, 2. doi:10.1002/aic.690190246

Frontini, M. & Sormani, E. 2003. Some variant of Newton’s method with third order convergence, Applied Mathematics and Computation, 140(2-3), pp. 419-426, doi:10.1016/S0096-3003(02)00238-2

Grabisch, M., Marichal, J. L., Mesiar, R., & Pap, E. 2009. Aggregation Function, Cambridge University Press.

Herceg, D. & Herceg, D. 2013a. Means based modifications of Newton’s method for solving nonlinear equations, Applied Mathematics and Computation, 219(11), pp. 6126-6133.

Herceg, D. & Herceg, D. 2013b. Third-order modifications of Newton’s method based on Stolarsky and Gini means, Journal of Computational and Applied Mathematics, 245, pp. 53-61. doi:10.1016/j.cam.2012.12.008

Homeier, H. H. H. 2003. A modified Newton method for rootfinding with cubic convergence, J. Comput. Appl. Math., 157(1), pp. 53-61. doi:10.1016/S0377-0427(03)00391-1

Jain, D. 2013. Families of Newton-like methods with fourthorder convergence, International Journal of computer mathematics, 90(5), pp. 1072-1082. doi:10.1080/00207160.2012.746677

Klement, E. P., Mesiar, R., & Pap, E. 2000. Triangular Norms, Kluwer Academic Publishers, Dordrecht. doi:10.1007/978-94-015-950-7

Klir, G. J. & Yuan, B. 1995. Fuzzy sets and fuzzy logic, Theory and applications, Prentice Hall, New Jersey.

Lukić, T. & Ralević, N. M. 2008. Geometric Mean Newton’s Method for Simple and Multiple Roots, Applied Mathematics Letters, 21, pp. 30-36. doi:10.1016/j.aml.2007.02.010

Lukić, T., Ralević, N. M., & Lukity, A. 2006, Application of Aggregation Operators in Solution of Nonlinear Equations, 4th Serbian-Hungarian Joint Symposium on Intelligent Systems, September 29-30, 2006, Subotica, Serbia and Montenegro, pp. 329-339.

Maroju, P., Behl, R., & Motsa, S. S. 2017. Some novel and optimal families of King’s method with eighth and sixteenth-order of convergence, Journal of Computational and Applied Mathematics, 318, pp. 136-148. doi:10.1016/j.cam.2016.11.018

McDougall, T. J. & Wotherspoon, S. J. 2014. A simple modification of Newton’s method to achieve convergence of order 1+2^(1/2), Applied Mathematics Letters, 29, pp. 20-25. doi:10.1016/j.aml.2013.10.008

Özban, A. Y. 2004. Some new variants of Newton’s method, Appl. Math. Lett., 17, pp. 677-682. doi:10.1016/S0893-9659(04)90104-8

Ralević, N. M. & Ćebić, D. 2019. The Newton Method for Solving Nonlinear Equations Based on Aggregation operator, SYM-OP-IS 2019, XLVI International Symposium on Operational Research, Kladovo, September 15-18, 2019, pp. 367-372.

Ralević, N. M. & Lukić, T. 2005. Modification of Newton’s Method Based on Root-Power Mean, Applied Linear Algebra 2005, October 13-15, Palic, Serbia and Montenegro.

Rudin, W. 1991. Functional analysis (McGraw-Hill) Wait, R. 1979. The Numerical Solution of Algebraic Equations, John Wiley & Sons.

Weerakoon, S. & Fernando, T. G. I. 2000. A variant of Newton’s method with accelarated third-order convergence, Appl. Math. Lett., 13(8), pp. 87-93. doi:10.1016/S0893-9659(00)00100-2

Zachary, J. L. 2012, Introduction to scientific programming: computational problem solving using Maple and C, Springer-Verlag New York, doi:10.1007/978-1-4612-2366-5