##
**Several new third-order and fourth-order iterative methods for solving nonlinear equations.**
*(English)*
Zbl 1334.65090

Summary: In order to find the zeros of nonlinear equations, in this paper, we propose a family of third-order and optimal fourth-order iterative methods. We have also obtained some particular cases of these methods. These methods are constructed through weight function concept. The multivariate case of these methods has also been discussed. The numerical results show that the proposed methods are more efficient than some existing third- and fourth-order methods.

### MSC:

65H05 | Numerical computation of solutions to single equations |

PDF
BibTeX
XML
Cite

\textit{A. Singh} and \textit{J. P. Jaiswal}, Int. J. Eng. Math. 2014, Article ID 828409, 11 p. (2014; Zbl 1334.65090)

Full Text:
DOI

### References:

[1] | S. Weerakoon and T. G. I. Fernando, “A variant of Newton/s method with accelerated third-order convergence,” Applied Mathematics Letters, vol. 13, no. 8, pp. 87-93, 2000. · Zbl 0973.65037 |

[2] | H. H. H. Homeier, “On Newton-type methods with cubic convergence,” Journal of Computational and Applied Mathematics, vol. 176, no. 2, pp. 425-432, 2005. · Zbl 1063.65037 |

[3] | C. Chun and Y. Kim, “Several new third-order iterative methods for solving nonlinear equations,” Acta Applicandae Mathematicae, vol. 109, no. 3, pp. 1053-1063, 2010. · Zbl 1195.41015 |

[4] | H. T. Kung and J. F. Traub, “Optimal order of one-point and multipoint iteration,” Journal of Computational and Applied Mathematics, vol. 21, no. 4, pp. 643-651, 1974. · Zbl 0289.65023 |

[5] | F. Soleymani, “Two new classes of optimal Jarratt-type fourth-order methods,” Applied Mathematics Letters, vol. 25, no. 5, pp. 847-853, 2012. · Zbl 1239.65030 |

[6] | M. Sharifi, D. K. R. Babajee, and F. Soleymani, “Finding the solution of nonlinear equations by a class of optimal methods,” Computers and Mathematics with Applications, vol. 63, no. 4, pp. 764-774, 2012. · Zbl 1247.65066 |

[7] | S. K. Khattri and S. Abbasbandy, “Optimal fourth order family of iterative methods,” Matematicki Vesnik, vol. 63, no. 1, pp. 67-72, 2011. · Zbl 1265.65094 |

[8] | W. Gautschi, Numerical Analysis: An Introduction, Birkhauser, Boston, Mass, USA, 1997. · Zbl 0877.65001 |

[9] | J. F. Traub, Iterative Methods for Solution of Equations, Chelsea Publishing, New York, NY, USA, 1997. · Zbl 0121.11204 |

[10] | M. Grau-Sánchez and J. L. Díaz-Barrero, “Zero-finder methods derived using Runge-Kutta techniques,” Applied Mathematics and Computation, vol. 217, no. 12, pp. 5366-5376, 2011. · Zbl 1229.65079 |

[11] | K. Huen, “Neue methode zur approximativen integration der differentialge-ichungen einer unabhngigen variablen,” Zeitschrift für angewandte Mathematik und Physik, vol. 45, pp. 23-38, 1900. |

[12] | F. Soleymani and D. K. R. Babajee, “Computing multiple zeros using a class of quartically convergent methods,” Alexandria Engineering Journal, vol. 52, pp. 531-541, 2013. |

[13] | M. A. Noor and M. Waseem, “Some iterative methods for solving a system of nonlinear equations,” Computers and Mathematics with Applications, vol. 57, no. 1, pp. 101-106, 2009. · Zbl 1165.65349 |

[14] | J. R. Sharma, R. K. Guha, and R. Sharma, “An efficient fourth-order weighted-Newton method for systems of nonlinear equations,” Numerical Algorithms, vol. 62, pp. 307-323, 2013. · Zbl 1283.65051 |

[15] | D. K. R. Babajee, A. Cordero, F. Soleymani, and J. R. Torregrosa, “On a novel fourth-order algorithm for solving systems of nonlinear equations,” Journal of Applied Mathematics, vol. 2012, Article ID 165452, 12 pages, 2012. · Zbl 1268.65072 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.