Abstract
In recent times, many iterative methods for computing multiple zeros of nonlinear functions have been appeared in literature. Different from these existing methods, here we propose a new class of methods with eighth order convergence for multiple zeros. With four evaluations per iteration, the methods satisfy the criterion of attaining optimal convergence of eighth order. Applicability is demonstrated on different examples that illustrates the computational efficiency of novel methods. Comparison of numerical results shows that the proposed techniques possess good convergence compared to existing optimal order techniques. Besides, the accuracy of existing techniques is also challenged which is the main advantage.
Similar content being viewed by others
References
Argyros, I.K., Regmi, S.: Undergraduate Research at Cameron University on Iterative Procedures in Banach and Other Spaces. Nova Science Publisher, New York (2019)
Dong, C.: A family of multiopoint iterative functions for finding multiple roots of equations. Int. J. Comput. Math. 21, 363–367 (1987)
Douglas, J.M.: Process Dynamics and Control. Prentice Hall, Englewood Cliffs (1972)
Hoffman, J.D.: Numerical Methods for Engineers and Scientists. McGraw-Hill Book Company, New York (1992)
Kansal, M., Kanwar, V., Bhatia, S.: On some optimal multiple root-finding methods and their dynamics. Appl. Appl. Math. 10, 349–367 (2015)
King, R.F.: A secant method for multiple roots. BIT 17, 321–328 (1977)
Kumar, S., Kumar, D., Sharma, J.R., Cesarano, C., Agarwal, P., Chu, Y.M.: An optimal fourth order derivative-free numerical algorithm for multiple roots. Symmetry 12, 1038 (2020). https://doi.org/10.3390/sym12061038
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)
Liu, B., Zhou, X.: A new family of fourth-order methods for multiple roots of nonlinear equations. Non. Anal. Model. Cont. 18, 143–152 (2013)
Ostrowski, A.M.: Solutions of Equations and System of Equations. Academic Press, New York (1960)
Schröder, E.: Über unendlich viele Algorithmen zur Auflösung 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)
Sharma, J.R., Kumar, D., Cattani, C.: An efficient class of weighted-Newton multiple root solvers with seventh order convergence. Symmetry 11, 1054 (2019). https://doi.org/10.3390/sym11081054
Sharma, J.R., Kumar, S., Jäntschi, L.: On derivative free multiple-root finders with optimal fourth order convergence. Mathematics 8, 1091 (2020). https://doi.org/10.3390/math8071091
Sharma, J.R., Sharma, R.: Modified Jarratt method for computing multiple roots. Appl. Math. Comput. 217, 878–881 (2010)
Soleymani, F., Babajee, D.K.R., Lotfi, T.: On a numerical technique for finding multiple zeros and its dynamics. J. Egypt. Math. Soc. 21, 346–353 (2013)
Traub, J.F.: Iterative Methods for the Solution of Equations. Chelsea Publishing Company, New York (1982)
Weerakoon, S., Fernando, T.G.I.: A variant of Newton’s method with accelerated third-order convergence. Appl. Math. Lett. 13, 87–93 (2000)
Wolfram, S.: Wolfram Mathematica, 12th edn. Wolfram Research, Champaign (2020)
Zhou, X., Chen, X., Song, Y.: Constructing higher-order methods for obtaining the multiple roots of nonlinear equations. J. Comput. Math. Appl. 235, 4199–4206 (2011)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Sharma, J.R., Kumar, S. A class of computationally efficient numerical algorithms for locating multiple zeros. Afr. Mat. 32, 853–864 (2021). https://doi.org/10.1007/s13370-020-00865-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13370-020-00865-3