Abstract
A number of higher order Newton-like methods (i.e. the methods requiring both function and derivative evaluations) are available in literature for multiple zeros of a nonlinear function. However, higher order Traub-Steffensen-like methods (i.e. the methods requiring only function evaluations) for computing multiple zeros are rare in literature. Traub-Steffensen-like iterations are important in the circumstances when derivatives are complicated to evaluate or expensive to compute. Motivated by this fact, here we present an efficient and rapid-converging Traub-Steffensen-like algorithm to locate multiple zeros. The method achieves eighth order convergence by using only four function evaluations per iteration, therefore, this convergence rate is optimal. Performance is demonstrated by applying the method on different problems including some real life models. The computed results are compared with that of existing optimal eighth-order Newton-like techniques to reveal the computational efficiency of the new approach.
Similar content being viewed by others
References
Akram, S., Zafar, F., Yasmin, N.: An optimal eighth-order family of iterative methods for multiple roots. Mathermatics (2019). https://doi.org/10.3390/math7080672
Argyros, I.K.: Convergence and Applications of Newton-Type Iterations. Springer-Verlag, New York, NY, USA (2008)
Argyros, I.K., Magreñán, Á.A.: Iterative Methods and Their Dynamics with Applications. CRC Press, New York, NY, USA (2017)
Behl, R., Alshomrani, A.S., Motsa, S.S.: An optimal scheme for multiple roots of nonlinear equations with eighth-order convergence. J. Math. Chem. 56, 2069–2084 (2018)
Behl, R., Cordero, A., Motsa, S.S., Torregrosa, J.R.: An eighth-order family of optimal multiple root finders and its dynamics. Numer. Algor. 77, 1249–1272 (2018)
Behl, R., Zafar, F., Alshormani, A.S., Junjua, M.U.D., Yasmin, N.: An optimal eighth-order scheme for multiple zeros of unvariate functions. Int. J. Comput. Meth. (2018). https://doi.org/10.1142/S0219876218430028
Chapra, S.C., Canale, R.P.: Numerical Methods for Engineers. McGraw-Hill Book Company, New York (1988)
Constantinides, A., Mostoufi, N.: Numerical Methods for Chemical Engineers with MATLAB Applications. Upper Saddle River, Prentice Hall PTR (1999)
Dong, C.: A family of multipoint 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)
Geum, Y.H., Kim, Y.I., Neta, B.: A class of two-point sixth-order multiplezero finders of modified double-Newton type and their dynamics. Appl. Math. Comput. 270, 387–400 (2015)
Geum, Y.H., Kim, Y.I., Neta, B.: Constructing a family of optimal eighth-order modified Newton-type multiple-zero finders along with the dynamics behind their purely imaginary extraneous fixed points. J. Comp. Appl. Math. 333, 131–156 (2018)
Hansen, E., Patrick, M.: A family of root finding methods. Numer. Math. 27, 257–269 (1977)
Hoffman, J.D.: Numerical Methods for Engineers and Scientists. McGraw-Hill Book Company, New York, NY, USA (1992)
Kung, H.T., Traub, J.F.: Optimal order of one-point and multipoint iteration. J. Assoc. Comput. Mach. 21, 643–651 (1974)
Osada, N.: An optimal multiple root-finding method of order three. J. Comput. Appl. Math. 51, 131–133 (1994)
Schröder, E.: Über unendlich viele Algorithmen zur Auflösung der Gleichungen. Math. Ann. 2, 317–365 (1870)
Sharma, J.R., Kumar, S., Jäntschi, L.: On a class of optimal fourth order multiple root solvers without using derivatives. Symmetry 11(1452), 1–14 (2019)
Sharma, J.R., Sharma, R.: Modified Jarratt method for computing multiple roots. Appl. Math. Comput. 217, 878–881 (2010)
Traub, J.F.: Iterative Methods for the Solution of Equations. Chelsea Publishing Company, New York, NY, USA (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.: The Mathematica Book, 5th edn. Wolfram Media, Champaign, IL, USA (2003)
Zafar, F., Cordero, A., Quratulain, R., Torregrosa, J.R.: Optimal iterative methods for finding multiple roots of nonlinear equations using free parameters. J. Math. Chem. 56, 1884–1901 (2017)
Zafar, F., Cordero, A., Sultana, S., Torregrosa, J.R.: Optimal iterative methods for finding multiple roots of nonlinear equations using weight functions and dynamics. J. Comp. Appl. Math. 342, 352–374 (2018)
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. An excellent derivative-free multiple-zero finding numerical technique of optimal eighth order convergence. Ann Univ Ferrara 68, 161–186 (2022). https://doi.org/10.1007/s11565-022-00394-w
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11565-022-00394-w