Basins of attraction for family of Popovski’s methods and their extension to multiple roots

Beny Neta

doi:10.33993/jnaat511-1248

Authors

Beny Neta Naval Postgraduate School, Monterey, USA

DOI:

https://doi.org/10.33993/jnaat511-1248

Keywords:

Nonlinear equations, simple roots, multiple roots, basins of attraction, Popovski's method

Abstract views: 271

Abstract

In this paper we revisit Popovski’s family of methods for simple roots. We compare several members using basins of attraction visually and qualitatively by comparing the run-time on several examples, the average number of iterations and the number of divergent points. We chose 5 different members of the family. We also develop an equivalent family of methods for multiple roots and compare several members on six different numerical examples.

Downloads

Download data is not yet available.

References

J. F. Traub, Iterative methods for the solution of equations, Prentice-Hall, Englewood Cliffs, 1964.

M. Petkovic, B. Neta, L. Petkovic, J. Dzunic, Multipoint Methods for Solving Nonlinear Equations. Academic Press, Boston, 2013 https://doi.org/10.1016/j.amc.2013.10.072 DOI: https://doi.org/10.1016/j.amc.2013.10.072

D. B. Popovski, A family of one point iteration formulae for finding roots, Int. J.Comput. Math. 8 (1980), 85-88. https://doi.org/10.1080/00207168008803193 DOI: https://doi.org/10.1080/00207168008803193

E. Halley, A new, exact and easy method of finding the roots of equations generally and that without any previous reduction, Phil. Trans. Roy. Soc. London 18 (1694) 136-148.http://dx.doi.org/10.1098/rstl.1694.0029 DOI: https://doi.org/10.1098/rstl.1694.0029

A. Cauchy, Sur la Determination Approximative des Racines d’une Equation Algebrique ou Transcendante, Oeuvres Completes Serie 2, Tome 4, Gauthier Villars, Paris, 1899, pp. 573-609.

S. Hitotumatu, A method of successive approximation based on the expansion of second order, Math. Japan, 7 (1962), 31-50.

B. Neta, On Popovski’s method for nonlinear equations, Applied. Math. Computer.201(2008) 710-715 https://doi.org/10.1016/j.amc.2008.01.012 DOI: https://doi.org/10.1016/j.amc.2008.01.012

M. Scott, B. Neta, C. Chun, Basin attractors for various methods, Appl. Math. Comput. 218 (2011), 2584-2599. https://doi.org/10.1016/j.amc.2011.07.076 DOI: https://doi.org/10.1016/j.amc.2011.07.076

B. Neta, M. Scott, C. Chun, Basins of attraction for several methods to find simple roots of nonlinear equations, Appl. Math. Comput. 218 (2012), 10548-10556. https://doi.org/10.1016/j.amc.2012.04.017 DOI: https://doi.org/10.1016/j.amc.2012.04.017

B. Neta, C. Chun, On a family of Halley-like methods to find simple roots of nonlinear equations, Appl. Math. Comput. 219 (2013), 7940-7944. https://doi.org/10.1016/j.amc.2013.02.035 DOI: https://doi.org/10.1016/j.amc.2013.02.035

C. Chun, B. Neta, Comparative study of methods of various orders for finding simple roots of nonlinear equations, J. Appl. Anal. Comput. 9 (2019), 400-427.https://doi.org/10.11948/2156-907X.2016022 DOI: https://doi.org/10.11948/2156-907X.20160229

E.R. Vrscay, W.J. Gilbert, Extraneous fixed points, basin boundaries and chaotic dynamics for Schroder and Konig rational iteration functions, Numer. Math. 52 (1988) 1-16. https://doi.org/10.1007/BF01401018 DOI: https://doi.org/10.1007/BF01401018

D. Herceg, I. Petkovic, Computer visualization and dynamic study of new families of root-solvers, J. Comput. Appl. Math., to appear. url-https://doi.org/10.1016/j.cam.2021.113775 DOI: https://doi.org/10.1016/j.cam.2021.113775

I. Petkovic, B. Neta, On an application of symbolic computation and computer graphics to root-finders: The case of multiple roots of unknown multiplicity, J. Comput. Appl. Math., 308(2016), 215-230. https://doi.org/10.1016/j.cam.2016.06.008 DOI: https://doi.org/10.1016/j.cam.2016.06.008

C. Chun, B. Neta, Basins of attraction for several third order methods to find multiple roots of nonlinear equations, Appl. Math. Comput., 268(2015), 129-137. https://doi.org/10.1016/j.amc.2015.06.068 DOI: https://doi.org/10.1016/j.amc.2015.06.068

B. Neta, C. Chun, On a family of Laguerre methods to find multiple roots of nonlinear equations, Appl. Math. Comput. 219 (2013), 10987-11004. https://doi.org/10.1016/j.amc.2013.05.002 DOI: https://doi.org/10.1016/j.amc.2013.05.002

S. Amat, S. Busquier, S. Plaza, Dynamics of a family of third-order iterative methods that do not require using second derivatives, Appl. Math. Comput., 154, (2004), 735-746. https://doi.org/10.1016/S0096-3003(03)00747-1 DOI: https://doi.org/10.1016/S0096-3003(03)00747-1

A. F. Beardon, Iteration of Rational Functions Complex Analytic Dynamical Systems, Springer-Verlag, New York, 1991. DOI: https://doi.org/10.1007/978-1-4612-4422-6

O. Passon, J. Grebe-Ellis, Planck’s radiation law, the light quantum, and the pre-history of indistinguishability in the teaching of quantum mechanics, Eur. J. Phys., 3(2017) 035404. https://doi.org/10.1088/1361-6404/aa6134 DOI: https://doi.org/10.1088/1361-6404/aa6134

R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, D. E. Knuth, On the LambertW Function, Advances in Computational Mathematics, Vol. 5 (1996), 329-359. DOI: https://doi.org/10.1007/BF02124750