# A robust cubically and quartically iterative techniques free from derivative

## DOI:

https://doi.org/10.4067/S0716-09172011000200002## Keywords:

Derivative-free methods, Efficiency index, Error equation, Asymptotic error constant, Multi-point iterations, Optimal order.## Abstract

Constructing of a technique which is both accurate and derivative-free is one of the most important tasks in the field of iterative processes. Hence in this study, convergent iterative techniques are suggested for solving single variable nonlinear equations. Their error equations are given theoretically to show that they have cubic and quartical convergence. Per iteration the novel schemes include three evaluations of the function while they are free from derivative as well. In viewpoint of optimality, the developed quartically class reaches the optimal efficiency index 41/3 ? 1.587 based on the Kung-Traub Hypothesis regarding the optimality of multi-point iterations without memory. In the end, the theoretical results are supported by numerical examples to elucidate the accuracy ofthe developed schemes.## References

[1] I. K. Argyros, S. Hilout, On the local convergence of a two-step Steffensen-type method for solving generalized equations, Proyecciones, 27, pp. 319-330 (2008).

[2] M. Dehghan, M. Hajarian, Some derivative free quadratic and cubic convergence iterative formulas for solving nonlinear equations, Comp. Appl. Math., 29, pp. 19-31 (2010).

[3] A. Iliev, N. Kyurkchiev, Q. Fang, On a generalization of the EulerChebyshev method for simultaneous extraction of only a part of all roots of polynomials, Japan J. Indust. Appl. Math., 23, pp. 63-73 (2006).

[4] A. Galantai, C. J. Hegedus, A study of accelerated Newton methods for multiple polynomial roots, Numer. Algorithms, 54, pp. 219-243 (2010).

[5] H. T. Kung, J.F. Traub, Optimal order of one-point and multipoint iteration, J. Assoc. Comput. Math., 21, pp. 643-651 (1974).

[6] Z. Liu, Q. Zheng, P. Zhao, A variant of Steffenson’s method of fourthorder convergence and its applications, Appl. Math. Comput., 216, pp. 1978-1983 (2010).

[7] A. Sidi, Generalization of the secant method for nonlinear equations, Appl. Math. E-Notes, 8, pp. 115-123 (2008).

[8] F. Soleymani, Revisit of Jarratt method for solving nonlinear equations, Numer. Algorithms (2010) doi: 10.1007/s11075-010-9433-6.

[9] F. Soleymani, V. Hosseinabadi, New third- and sixth-order derivativefree techniques for nonlinear equations, J. Math. Res., 3, pp. 107-112, (2011).

[10] F. Soleymani, M. Sharifi, A simple but efficient multiple zero-finder for solving nonlinear equations, Far East J. Math. Sci. (FJMS), 42, pp. 153-160, (2010).

[11] F. Soleymani, Concerning some sixth-order iterative methods for finding the simple roots of nonlinear equations, Bull. Math. Anal. Appl., 2, pp. 146-151, (2010).

[12] F. Soleymani, Regarding the accuracy of optimal eighth-order methods, Math. Comput. Modelling, 53, pp. 1351- 1357, (2011).

[13] F. Soleimani, F. Soleymani, Computing simple roots by a sixth-order iterative method, Int. J. Pure Appl. Math., 69, pp. 41-48, (2011).

[14] P. Sargolzaei, F. Soleymani, Accurate fourteenth-order methods for solving nonlinear equations, Numer. Algorithms, (2011), doi: 10.1007/s11075-011-9467-4.

[15] J. F. Steffensen, Remarks on iteration, Skand. Aktuarietidskr, 16, pp. 64-72, (1933).

[2] M. Dehghan, M. Hajarian, Some derivative free quadratic and cubic convergence iterative formulas for solving nonlinear equations, Comp. Appl. Math., 29, pp. 19-31 (2010).

[3] A. Iliev, N. Kyurkchiev, Q. Fang, On a generalization of the EulerChebyshev method for simultaneous extraction of only a part of all roots of polynomials, Japan J. Indust. Appl. Math., 23, pp. 63-73 (2006).

[4] A. Galantai, C. J. Hegedus, A study of accelerated Newton methods for multiple polynomial roots, Numer. Algorithms, 54, pp. 219-243 (2010).

[5] H. T. Kung, J.F. Traub, Optimal order of one-point and multipoint iteration, J. Assoc. Comput. Math., 21, pp. 643-651 (1974).

[6] Z. Liu, Q. Zheng, P. Zhao, A variant of Steffenson’s method of fourthorder convergence and its applications, Appl. Math. Comput., 216, pp. 1978-1983 (2010).

[7] A. Sidi, Generalization of the secant method for nonlinear equations, Appl. Math. E-Notes, 8, pp. 115-123 (2008).

[8] F. Soleymani, Revisit of Jarratt method for solving nonlinear equations, Numer. Algorithms (2010) doi: 10.1007/s11075-010-9433-6.

[9] F. Soleymani, V. Hosseinabadi, New third- and sixth-order derivativefree techniques for nonlinear equations, J. Math. Res., 3, pp. 107-112, (2011).

[10] F. Soleymani, M. Sharifi, A simple but efficient multiple zero-finder for solving nonlinear equations, Far East J. Math. Sci. (FJMS), 42, pp. 153-160, (2010).

[11] F. Soleymani, Concerning some sixth-order iterative methods for finding the simple roots of nonlinear equations, Bull. Math. Anal. Appl., 2, pp. 146-151, (2010).

[12] F. Soleymani, Regarding the accuracy of optimal eighth-order methods, Math. Comput. Modelling, 53, pp. 1351- 1357, (2011).

[13] F. Soleimani, F. Soleymani, Computing simple roots by a sixth-order iterative method, Int. J. Pure Appl. Math., 69, pp. 41-48, (2011).

[14] P. Sargolzaei, F. Soleymani, Accurate fourteenth-order methods for solving nonlinear equations, Numer. Algorithms, (2011), doi: 10.1007/s11075-011-9467-4.

[15] J. F. Steffensen, Remarks on iteration, Skand. Aktuarietidskr, 16, pp. 64-72, (1933).

## Published

2011-12-09

## How to Cite

[1]

F. Soleymani and V. Hosseinabadi, “A robust cubically and quartically iterative techniques free from derivative”,

*Proyecciones (Antofagasta, On line)*, vol. 30, no. 2, pp. 149-161, Dec. 2011.## Issue

## Section

Artículos