Calcolo

, Volume 53, Issue 2, pp 201–215

# Solving nonlinear equations by a derivative-free form of the King’s family with memory

• Somayeh Sharifi
• Stefan Siegmund
• Mehdi Salimi
Article

## Abstract

In this paper, we present an iterative three-point method with memory based on the family of King’s methods to solve nonlinear equations. This proposed method has eighth order convergence and costs only four function evaluations per iteration which supports the Kung-Traub conjecture on the optimal order of convergence. An acceleration of the convergence speed is achieved by an appropriate variation of a free parameter in each step. This self accelerator parameter is estimated using Newton’s interpolation polynomial of fourth degree. The order of convergence is increased from 8 to 12 without any extra function evaluation. Consequently, this method, possesses a high computational efficiency. Finally, a numerical comparison of the proposed method with related methods shows its effectiveness and performance in high precision computations.

## Keywords

Multi-point method Nonlinear equations Method with memory R-order of convergence Kung-Traub’s conjecture

65H05

## References

1. 1.
Chun, C., Lee, M.Y.: A new optimal eighth-order family of iterative methods for the solution of nonlinear equations. Appl. Math. Comput. 223, 506–519 (2013)
2. 2.
Cordero, A., Torregrosa, J.R., Vassileva, M.P.: Three-step iterative methods with optimal eighth-order convergence. J. Comput. Appl. Math. 235, 3189–3194 (2011)
3. 3.
Dzunic, J., Petkovic, M.S., Petkovic, L.D.: Three-point methods with and without memory for solving nonlinear equatins. Appl. Math. Comput. 218, 4917–4927 (2012)
4. 4.
Hazrat, R.: Mathematica: a problem-centered approach. Springer, New York (2010)
5. 5.
Jarratt, P.: Some fourth order multipoint iterative methods for solving equations. Math. Comput. 20, 434–437 (1966)
6. 6.
King, R.F.: Family of four order methods for nonlinear equations. SIAM. J. Numer. Anal. 10, 876–879 (1973)
7. 7.
Kung, H.T., Traub, J.F.: Optimal order of one-point and multipoint iteration. J. Assoc. Comput. Math. 21, 634–651 (1974)
8. 8.
Lotfi, T., Sharifi, S., Salimi, M., Siegmund, S.: A new class of three-point methods with optimal convergence order eight and its dynamics. Numer. Algor. 68, 261–288 (2015)
9. 9.
Neta, B.: On a family of multipoint methods for nonlinear equations. Int. J. Computer Math. 9, 353–361 (1981)
10. 10.
Ostrowski, A.M.: Solution of equations and systems of equations. Academic Press, New York (1966)
11. 11.
Petkovic, M.S., Dzunic, J., Neta, B.: Interpolatory multipoint methods with memory for solving nonlinear equations. Appl. Math. Comput. 218, 2533–2541 (2011)
12. 12.
Petkovic, M.S., Neta, B., Petkovic, L.D., Dzunic, J.: Multipoint methods for solving nonlinear equations. Elsevier, Waltham (2013)
13. 13.
Salimi, M., Lotfi, T., Sharifi, S., Siegmund, S.: Optimal newton-secant like methods without memory for solving nonlinear equations with its dynamics. Under Revision, (2015)Google Scholar
14. 14.
Sharifi, S. Ferrara, M. Salimi, M. Siegmund, S.: New modification of Maheshwari method with optimal eighth order of convergence for solving nonlinear equations. Under Revision, (2015)Google Scholar
15. 15.
Sharifi, S. Salimi, M. Siegmund, S. Lotfi, T.: A new class of optimal four-point methods with convergence order 16 for solving nonlinear equations. Under Revision, (2015)Google Scholar
16. 16.
Sharma, J.R., Guha, R.K., Gupa, P.: Some efficient derivative free methods with memory for solving nonlinear equations. Appl. Math. Comput. 219, 699–707 (2012)
17. 17.
Traub, J.F.: Iterative methods for the solution of equations. Prentice Hall, New York (1964)
18. 18.
Weerakoon, S., Fernando, T.G.I.: A variant of Newton’s method with accelerated third-order convergence. Appl. Math. Lett. 13, 87–93 (2000)
19. 19.
Zheng, Q., Li, J., Huang, F.: An optimal Steffensen-type family for solving nonlinear equations. Appl. Math. Comput. 217, 9592–9597 (2011)

## Authors and Affiliations

• Somayeh Sharifi
• 1
Email author
• Stefan Siegmund
• 2
• Mehdi Salimi
• 2
1. 1.Young Researchers and Elite Club, Hamedan BranchIslamic Azad UniversityHamedanIran
2. 2.Center for Dynamics, Department of MathematicsTechnische Universität DresdenDresdenGermany