# Local Convergence Results for an Optimal Iterative Method for Multiple Roots

• Ramandeep Behl
• Eulalia Martínez
• Fabricio Cevallos
• Ali Saleh Alshomrani
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11386)

## Abstract

In this paper our aim is to perform a local convergence study of a fourth order iterative method in the case of multiple roots. As far as we know, these kind of studies have only been performed for iterative methods of second and third order of convergence in the case of multiple roots. So it is our purpose to analyze the radius of local convergence for higher-order methods. Usually the local convergence radius decreases when the order of the method increases, so it is necessary to study its behavior when we propose a new iterative method. In this sense, we introduce in this paper a new idea for establishing local convergence results of iterative methods for locating multiple zeros, under the assumption of a bounding condition for the $$(m + 1)-th$$ derivative of the function f(x) in its existence domain. We apply this technique to the modification of the Maheshwari fourth order method for the case of multiple roots. Finally, we perform some numerical examples that confirm the theoretical results established in this paper.

## Notes

### Acknowledgements

Supported by the project of Generalitat Valenciana Prometeo/2016/089 and MTM2014-52016-C2-2-P of the Spanish Ministry of Science and Innovation.

## References

1. 1.
Traub, J.F.: Iterative Methods for the Solution of Equations. Prentice-Hall, Englewood Cliffs (1964)
2. 2.
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)
3. 3.
Neta, B.: Extension of Murakami’s high-order non-linear solver to multiple roots. Int. J. Comput. Math. 87(5), 1023–1031 (2010)
4. 4.
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)
5. 5.
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)
6. 6.
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)
7. 7.
Soleymani, F., Babajee, D.K.R.: Computing multiple zeros using a class of quartically convergent methods. Alex. Eng. J. 52, 531–541 (2013)
8. 8.
Soleymani, F., Babajee, D.K.R., Lofti, T.: On a numerical technique for finding multiple zeros and its dynamic. J. Egypt. Math. Soc. 21, 346–353 (2013)
9. 9.
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)
10. 10.
Hueso, J.L., Martínez, E., Teruel, C.: Determination of multiple roots of nonlinear equations and applications. J. Math. Chem. 53, 880–892 (2015)
11. 11.
Argyros, I.: On the convergence and application of Newton’s method under weak Hölder continuity assumptions. Int. J. Comput. Math. 80, 767–780 (2003)
12. 12.
Zhou, X., Chen, X., Song, Y.: On the convergence radius of the modified Newton method for multiple roots under the center-Hölder condition. Numer. Algorithms 65, 221–232 (2014)
13. 13.
Bi, W., Ren, H., Wu, Q.: Convergence of the modified Halley’s method for multiple zeros under Hölder continuous derivative. Numer. Algorithms 58, 497–512 (2011)
14. 14.
Zhou, X., Son, Y.: Convergence radius of Osada’s method under center-Hölder continuous condition. Appl. Math. Comput. 243, 809–816 (2014)
15. 15.
Maheshwari, A.K.: A fourth order iterative method for solving nonlinear equations. Appl. Math. Comput. 211(2), 383–391 (2009)
16. 16.
Cordero, A., Torregrosa, J.R.: Variants of Newton’s method using fifth-order quadrature formulas. Appl. Math. Comput. 190(1), 686–698 (2007)
17. 17.
Balaji, G.V., Seader, J.D.: Application of interval Newton’s method to chemical engineering problems. Rel. Comput. 1(3), 215–223 (1995)
18. 18.
Shacham, M.: An improved memory method for the solution of a nonlinear equation. Chem. Eng. Sci. 44(7), 1495–1501 (1989)

© Springer Nature Switzerland AG 2019

## Authors and Affiliations

• Ramandeep Behl
• 1
• Eulalia Martínez
• 2
Email author
• Fabricio Cevallos
• 3
• Ali Saleh Alshomrani
• 1
1. 1.Department of MathematicsKing Abdulaziz UniversityJeddahSaudi Arabia
2. 2.Instituto Universitario de Matemática MultidisciplinarUniversitat Politècnica de ValènciaValenciaSpain
3. 3.Facultad de Ciencias EconómicasUniversidad laica “Eloy Alfaro” de ManabíMantaEcuador