Abstract
There is a few number of optimal fourth-order iterative methods for obtaining the multiple roots of nonlinear equations. But, in most of the earlier studies, scholars gave the flexibility in their proposed schemes only at the second step (not at the first step) in order to explore new schemes. Unlike what happens in existing methods, the main aim of this manuscript is to construct a new fourth-order optimal scheme which will give the flexibility to the researchers at both steps as well as faster convergence, smaller residual errors and asymptotic error constants. The construction of the proposed scheme is based on the mid-point formula and weight function approach. From the computational point of view, the stability of the resulting class of iterative methods is studied by means of the conjugacy maps and the analysis of strange fixed points. Their basins of attractions and parameter planes are also given to show their dynamical behavior around the multiple roots. Finally, we consider a real-life problem and a concrete variety of standard test functions for numerical experiments and relevant results are extensively treated to confirm the theoretical development.
Similar content being viewed by others
References
Behl, R., Cordero, A., Motsa, S.S., Torregrosa, J.R., Kanwar, V.: An optimal fourth-order family of methods for multiple roots and its dynamics. Numer. Algorithms 71(4), 775–796 (2016)
Blanchard, P.: Complex analytic dynamics on the Riemann sphere. Bull. Am. Math. Soc. 11(1), 85–141 (1984)
Chicharro, F., Cordero, A., Torregrosa, J.R.: Drawing dynamical and parameter planes of iterative families and methods. Sci. World J. 2013(2013), 1–11 (2013)
Devaney, R.L.: The Mandelbrot Set, the Farey Tree and the Fibonacci sequence. Am. Math. Mon. 106(4), 289–302 (1999)
Dong, C.: A family of multipoint iterative functions for finding multiple roots of equations. Int. J. Comput. Math. 21, 363–367 (1987)
Hueso, J.L., Martínez, E., Teruel, C.: Determination of multiple roots of nonlinear equations and applications. J. Math. Chem. 53, 880–892 (2015)
Kung, H.T., Traub, J.F.: Optimal order of one-point and multipoint iteration. J. Assoc. Comput. Mach. 21, 643–651 (1974)
Li, S.G., Cheng, L.Z., Neta, B.: Some fourth-order nonlinear solvers with closed formulae for multiple roots. Comput. Math. Appl. 59, 126–135 (2010)
Li, S., Liao, X., Cheng, L.: A new fourth-order iterative method for finding multiple roots of nonlinear equations. Appl. Math. Comput. 215, 1288–1292 (2009)
Petković, M.S., Neta, B., Petković, L.D., Dz̆unić, J.: Multipoint Methods for Solving Nonlinear Equations. Academic Press, New York (2013)
Sbibih, D., Serghini, A., Tijini, A., Zidna, A.: A general family of third order method for finding multiple roots. AMC 233, 338–350 (2014)
Schröder, E.: Über unendlichviele Algorithm zur Auffosung der Gleichungen. Math. Ann. 2, 317–365 (1870)
Sharifi, M., Babajee, D.K.R., Soleymani, F.: Finding the solution of nonlinear equations by a class of optimal methods. Comput. Math. Appl. 63, 764–774 (2012)
Soleymani, F., Babajee, D.K.R.: Computing multiple zeros using a class of quartically convergent methods. Alex. Eng. J. 52, 531–541 (2013)
Soleymani, F., Babajee, D.K.R., Lofti, T.: On a numerical technique forfinding multiple zeros and its dynamic. J. Egypt. Math. Soc. 21, 346–353 (2013)
Traub, J.F.: Iterative Methods for the Solution of Equations. Prentice-Hall, Englewood Cliffs (1964)
Zhou, X., Chen, X., Song, Y.: Families of third and fourth order methods for multiple roots of nonlinear equations. Appl. Math. Comput. 219, 6030–6038 (2013)
Zhou, X., Chen, X., Song, Y.: Constructing higher-order methods for obtaining the muliplte roots of nonlinear equations. J. Comput. Math. Appl. 235, 4199–4206 (2011)
Acknowledgements
The authors would like to thank the anonymous reviewers for their suggestions that have improved the final version of this manuscript.
Author information
Authors and Affiliations
Corresponding author
Additional information
This research was partially supported by Ministerio de Economía y Competitividad MTM2014-52016-C2-2-P and Generalitat Valenciana PROMETEO/2016/089.
Rights and permissions
About this article
Cite this article
Behl, R., Cordero, A., Motsa, S.S. et al. Multiplicity anomalies of an optimal fourth-order class of iterative methods for solving nonlinear equations. Nonlinear Dyn 91, 81–112 (2018). https://doi.org/10.1007/s11071-017-3858-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11071-017-3858-6