Numerical Algorithms

, Volume 71, Issue 4, pp 775–796

# An optimal fourth-order family of methods for multiple roots and its dynamics

Original Paper

## Abstract

There are few optimal fourth-order methods for solving nonlinear equations when the multiplicity m of the required root is known in advance. Therefore, the principle focus of this paper is on developing a new fourth-order optimal family of iterative methods. From the computational point of view, the conjugacy maps and the strange fixed points of some iterative methods are discussed, their basins of attractions are also given to show their dynamical behavior around the multiple roots. Further, using Mathematica with its high precision compatibility, a variety of concrete numerical experiments and relevant results are extensively treated to confirm the theoretical development.

## Keywords

Nonlinear equations Multiple roots Chebyshev’s method Schröder method Basin of attraction Complex dynamics

## References

1. 1.
Blanchard, P.: Complex analytic dynamics on the Riemann sphere. Bull. AMS 11(1), 85–141 (1984)
2. 2.
Blanchard, P.: The dynamics of Newton’s method. Proc. Symp. Appl. Math. 49, 139–154 (1994)
3. 3.
Chicharro, F., Cordero, A., Torregrosa, J.R.: Drawing dynamical and parameter planes of iterative families and methods. Sci. World J. 2013 (2013). Article ID 780153Google Scholar
4. 4.
Devaney, R.L.: The mandelbrot set, the farey tree and the fibonacci sequence. Amer. Math. Monthly 106(4), 289–302 (1999)
5. 5.
Kung, H.T., Traub, J.F.: Optimal order of one-point and multipoint iteration. J. Assoc. Comput. Mach. 21, 643–651 (1974)
6. 6.
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)
7. 7.
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)
8. 8.
Neta, B., Scott, M., Chun, C.: Basins attractors for various methods for multiple roots. Appl. Math. Comput. 218, 5043–5066 (2012)
9. 9.
Petkovic, M.S., Neta, B., Petkovic, L.D., Dzunic, J.: Multipoint methods for solving nonlinear equations. Academic Press (2013)Google Scholar
10. 10.
Sharma, J.R., Sharma, R.: Modified Jarratt method for computing multiple roots. Appl. Math. Comput. 217, 878–881 (2010)
11. 11.
Schröder, E.: Über unendlichviele Algorithm zur Auffosung der Gleichungen. Math. Annal. 2, 317–365 (1870)
12. 12.
Scott, M., Neta, B., Chun, C.: Basins attractors for various methods. Appl. Math. Comput. 218, 2584–2599 (2011)
13. 13.
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)
14. 14.
Traub, J.F.: Iterative Methods for the solution of equations. Prentice-Hall, Englewood Cliffs (1964)
15. 15.
Weerakoon, S., Fernando, T.G.I.: A variant of Newton’s method with accelerated third-order convergence. Appl. Math. Lett. 13, 87–93 (2000)
16. 16.
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)

## Authors and Affiliations

• Ramandeep Behl
• 1
• Alicia Cordero
• 2
• Sandile S. Motsa
• 1
• Juan R. Torregrosa
• 2
Email author
• Vinay Kanwar
• 3
1. 1.School of Mathematics, Statistics and Computer ScienceUniversity of KwaZulu-NatalPietermaritzburgSouth Africa
2. 2.Instituto de Matemática MultidisciplinarUniversitat Politècnica de ValènciaValenciaSpain
3. 3.University Institute of Engineering and TechnologyPanjab UniversityChandigarhIndia