A Family of Methods for Solving Nonlinear Equations with Twelfth-Order Convergence

Affiliation(s)

Department of Mathematics and Statistics, Qinghai University for Nationalities, Xining, China.

Department of Mathematics and Statistics, Qinghai University for Nationalities, Xining, China.

ABSTRACT

This paper presents a new family of twelfth-order methods for solving simple roots of nonlinear equations which greatly improves the order of convergence and the computational efficiency of the Newton’s method and some other known methods.

Cite this paper

X. Liu and X. Wang, "A Family of Methods for Solving Nonlinear Equations with Twelfth-Order Convergence,"*Applied Mathematics*, Vol. 4 No. 2, 2013, pp. 326-329. doi: 10.4236/am.2013.42049.

X. Liu and X. Wang, "A Family of Methods for Solving Nonlinear Equations with Twelfth-Order Convergence,"

References

[1] J. F. Traub, “Iterative Methods for Solutions of Equations,” Prentice-Hall, Upper Saddle River, 1964.

[2] C. Chun, “Some Third-Order Families of Iterative Methods for Solving Nonlinear Equations,” Applied Mathematics and Computation, Vol. 188, No. 1, 2007, pp. 924-933. doi:10.1016/j.amc.2006.09.113

[3] C. Chun and Y. M. Ham, “Some Second-Derivative-Free of Super-Halley Method with Fourth-Order Convergence,” Applied Mathematics and Computation, Vol. 195, No. 2, 2008, pp. 532-541. doi:10.1016/j.amc.2007.05.003

[4] M. T. Darvish and A. Barati, “A Third-Order NewtonType Method to Solve Systems of Nonlinear Equations,” Applied Mathematics and Computation, Vol. 187, No. 2, 2007, pp. 630-635. doi:10.1016/j.amc.2006.08.080

[5] M. T. Darvish, “A Two-Step High-Order Newton-Like Method to Solve Systems of Nonlinear Equations,” International Journal of Pure and Applied Mathematics, Vol. 57, No. 4, 2009, pp. 543-555.

[6] M. T. Darvishi, “Some Three-Step Iterative Methods Free from Second Order Derivative for Finding Solutions of Systems of Nonlinear Equations,” International Journal of Pure and Applied Mathematics, Vol. 57, No. 4, 2009, pp. 557-573.

[7] M. Frontini and E. Sormani, “Third-Order Methods from Quadrature Formulae for Solving Systems of Nonlinear Equations,” Applied Mathematics and Computation, Vol. 149, No. 3, 2004, pp. 771-782. doi:10.1016/S0096-3003(03)00178-4

[8] H. H. H. Homeier, “On Newton-Type Methods for Multiple Roots with Cubic Convergence,” Journal of Computational and Applied Mathematics, Vol. 231, No. 1, 2009, pp. 249-254. doi:10.1016/j.cam.2009.02.006

[9] J. Kou, Y. Li and X. Wang, “An Improvement of the Jarratt Method,” Applied Mathematics and Computation, Vol. 189, No. 2, 2007, pp. 1816-1821. doi:10.1016/j.amc.2006.12.062

[10] Y.-I. Kim and C. Chun, “New Twelfth-Order Modifications of Jarratt’s Method for Solving Nonlinear Equations,” Studies in Nonlinear Sciences, Vol. 1, No. 1, 2010, pp. 14-18.

[11] F. Liang, G. P. He and Z. Y. Hu, “A Cubically Convergent Newton-Type Method under Weak Conditions,” Journal of Computational and Applied Mathematics, Vol. 220, No. 1-2, 2008, pp. 409-412. doi:10.1016/j.cam.2007.08.013

[12] M. A. Noor and K. I. Noor, “Modified Iterative Methods with Cubic Convergence for Solving Nonlinear Equations,” Applied Mathematics and Computation, Vol. 184, No. 2, 2007, pp. 322-325. doi:10.1016/j.amc.2006.05.155

[13] J. R. Sharma, R. K. Guha and R. Sharma, “Improved Ostrowski-Like Methods Based on Cubic Curve Interpolation,” Applied Mathematics, Vol. 2, No. 7, 2011, pp. 816-823. doi:10.4236/am.2011.27109

[14] S. Weerakoon and T. G. I. Fernando, “A Variant of Newton’s Method with Accelerated Third-Order Convergence,” Applied Mathematics Letter, Vol. 13, No. 8, 2000, pp. 87-93. doi:10.1016/S0893-9659(00)00100-2

[1] J. F. Traub, “Iterative Methods for Solutions of Equations,” Prentice-Hall, Upper Saddle River, 1964.

[2] C. Chun, “Some Third-Order Families of Iterative Methods for Solving Nonlinear Equations,” Applied Mathematics and Computation, Vol. 188, No. 1, 2007, pp. 924-933. doi:10.1016/j.amc.2006.09.113

[3] C. Chun and Y. M. Ham, “Some Second-Derivative-Free of Super-Halley Method with Fourth-Order Convergence,” Applied Mathematics and Computation, Vol. 195, No. 2, 2008, pp. 532-541. doi:10.1016/j.amc.2007.05.003

[4] M. T. Darvish and A. Barati, “A Third-Order NewtonType Method to Solve Systems of Nonlinear Equations,” Applied Mathematics and Computation, Vol. 187, No. 2, 2007, pp. 630-635. doi:10.1016/j.amc.2006.08.080

[5] M. T. Darvish, “A Two-Step High-Order Newton-Like Method to Solve Systems of Nonlinear Equations,” International Journal of Pure and Applied Mathematics, Vol. 57, No. 4, 2009, pp. 543-555.

[6] M. T. Darvishi, “Some Three-Step Iterative Methods Free from Second Order Derivative for Finding Solutions of Systems of Nonlinear Equations,” International Journal of Pure and Applied Mathematics, Vol. 57, No. 4, 2009, pp. 557-573.

[7] M. Frontini and E. Sormani, “Third-Order Methods from Quadrature Formulae for Solving Systems of Nonlinear Equations,” Applied Mathematics and Computation, Vol. 149, No. 3, 2004, pp. 771-782. doi:10.1016/S0096-3003(03)00178-4

[8] H. H. H. Homeier, “On Newton-Type Methods for Multiple Roots with Cubic Convergence,” Journal of Computational and Applied Mathematics, Vol. 231, No. 1, 2009, pp. 249-254. doi:10.1016/j.cam.2009.02.006

[9] J. Kou, Y. Li and X. Wang, “An Improvement of the Jarratt Method,” Applied Mathematics and Computation, Vol. 189, No. 2, 2007, pp. 1816-1821. doi:10.1016/j.amc.2006.12.062

[10] Y.-I. Kim and C. Chun, “New Twelfth-Order Modifications of Jarratt’s Method for Solving Nonlinear Equations,” Studies in Nonlinear Sciences, Vol. 1, No. 1, 2010, pp. 14-18.

[11] F. Liang, G. P. He and Z. Y. Hu, “A Cubically Convergent Newton-Type Method under Weak Conditions,” Journal of Computational and Applied Mathematics, Vol. 220, No. 1-2, 2008, pp. 409-412. doi:10.1016/j.cam.2007.08.013

[12] M. A. Noor and K. I. Noor, “Modified Iterative Methods with Cubic Convergence for Solving Nonlinear Equations,” Applied Mathematics and Computation, Vol. 184, No. 2, 2007, pp. 322-325. doi:10.1016/j.amc.2006.05.155

[13] J. R. Sharma, R. K. Guha and R. Sharma, “Improved Ostrowski-Like Methods Based on Cubic Curve Interpolation,” Applied Mathematics, Vol. 2, No. 7, 2011, pp. 816-823. doi:10.4236/am.2011.27109

[14] S. Weerakoon and T. G. I. Fernando, “A Variant of Newton’s Method with Accelerated Third-Order Convergence,” Applied Mathematics Letter, Vol. 13, No. 8, 2000, pp. 87-93. doi:10.1016/S0893-9659(00)00100-2