# Power means based modification of Newton’s method for solving nonlinear equations with cubic convergence

### Abstract

In this paper, a class of Newton-type methods known as trapezoidal power means Newton method for solving nonlinear equation is proposed. The new methods incorporate power means in the trapezoidal integration rule along with midpoint, thus replacing in the classical Newton method. Some known variants can be regarded as particular cases of this method. The order of convergence of these methods is shown to be three. Numerical examples and their results are provided to compare the efficiency of the new methods with few other similar methods.### References

[1] D.K.R. Babajee, M.Z. Dauhoo, An analysis of the properties of the variants of Newton’s method with third order convergence, Appl. Math. Comput. 183 (2006) 659-684.

[2] A. Cordero, J. R. Torregrosa, Variants of Newton’s Method using fifth-order quadrature formulas, Applied Mathematics and Computation 190 (2007) 686-698.

[3] M. Frontini, E. Sormoni, Some variants of Newton’s method with third order convergence, Appl. Math. Comput. 140 (2003) 419-426.

[4] V. I. Hasanov, I. G. Ivanov, G. Nedjibov, A New Modification of Newton’s Method, Applications of Mathematics in Engineering 27 (2002) 278 -286.

[5] D. Herceg, Dj. Herceg, Means based modifications of Newton’s method for solving nonlinear equations, Appl. Math. Comput. 219 (2013) 6126-6133.

[6] H.H.H. Homeier, On Newton type methods with cubic convergence, J. Comput. Appl. Math. 176 (2005) 425-432.

[7] J. Jayakumar, M. Kalyanasundaram, A class of Newton's method using Simpson's rule having cubic and fifth order convergence, Appl. Math. Comput. (2013) (in press).

[8] J. Kou, Y. Li, X.Wang, Third-order modification of Newton’s method, J. Comput. Appl. Math. 205 (2007) 1 – 5.

[9] T. Lukic, N.M. Ralevic, Geometric mean Newton’s method for simple and multiple roots, Appl. Math. Lett. 21 (2008) 30-36.

[10] G. Nedzhibov, On a few iterative methods for solving nonlinear equations. Application of Mathematics in Engineering and Economics’28, in: Proceeding of the XXVIII Summer school Sozopol’ 02, pp.1-8, Heron press, Sofia, 2002.

[11] A.Y. Ozban, Some New Variants of Newton’s Method, Appl. Math. Lett. 17 (2004) 677-682.

[12] S. Weerakoon. T.G.I. Fernando, A Variant of Newton’s method with accelerated third-order convergence, Appl. Math. Lett. 13 (8), (2000) 87-93.

[13] Z. Xiaojian, A class of Newton’s methods with third-order convergence, Appl. Math. Lett. 20 (2007) 1026 – 1030.

[14] F. Zafar, N.A. Mir, A generalized family of quadrature based iterative methods, General Mathematics 18(4) (2010) 43-51.

[2] A. Cordero, J. R. Torregrosa, Variants of Newton’s Method using fifth-order quadrature formulas, Applied Mathematics and Computation 190 (2007) 686-698.

[3] M. Frontini, E. Sormoni, Some variants of Newton’s method with third order convergence, Appl. Math. Comput. 140 (2003) 419-426.

[4] V. I. Hasanov, I. G. Ivanov, G. Nedjibov, A New Modification of Newton’s Method, Applications of Mathematics in Engineering 27 (2002) 278 -286.

[5] D. Herceg, Dj. Herceg, Means based modifications of Newton’s method for solving nonlinear equations, Appl. Math. Comput. 219 (2013) 6126-6133.

[6] H.H.H. Homeier, On Newton type methods with cubic convergence, J. Comput. Appl. Math. 176 (2005) 425-432.

[7] J. Jayakumar, M. Kalyanasundaram, A class of Newton's method using Simpson's rule having cubic and fifth order convergence, Appl. Math. Comput. (2013) (in press).

[8] J. Kou, Y. Li, X.Wang, Third-order modification of Newton’s method, J. Comput. Appl. Math. 205 (2007) 1 – 5.

[9] T. Lukic, N.M. Ralevic, Geometric mean Newton’s method for simple and multiple roots, Appl. Math. Lett. 21 (2008) 30-36.

[10] G. Nedzhibov, On a few iterative methods for solving nonlinear equations. Application of Mathematics in Engineering and Economics’28, in: Proceeding of the XXVIII Summer school Sozopol’ 02, pp.1-8, Heron press, Sofia, 2002.

[11] A.Y. Ozban, Some New Variants of Newton’s Method, Appl. Math. Lett. 17 (2004) 677-682.

[12] S. Weerakoon. T.G.I. Fernando, A Variant of Newton’s method with accelerated third-order convergence, Appl. Math. Lett. 13 (8), (2000) 87-93.

[13] Z. Xiaojian, A class of Newton’s methods with third-order convergence, Appl. Math. Lett. 20 (2007) 1026 – 1030.

[14] F. Zafar, N.A. Mir, A generalized family of quadrature based iterative methods, General Mathematics 18(4) (2010) 43-51.

Published

2015-01-28

How to Cite

.
Power means based modification of Newton’s method for solving nonlinear equations with cubic convergence.

**International Journal of Applied Mathematics and Computation**, India, v. 6, n. 2, p. 1-6, jan. 2015. ISSN 0974-4673. Available at: <http://www.darbose.in/ojs/index.php/ijamc/article/view/631>. Date accessed: 13 dec. 2018. doi: https://doi.org/10.0000/ijamc.2014.6.2.631.
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Section

Articles

### Keywords

Non-linear equation; Iterative Methods; Newton’s Method; Rate of convergence; power mean

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