# Improved Newton-Raphson Methods for Solving Nonlinear Equations

## Main Article Content

## Abstract

In this paper, we mainly study the numerical algorithms for simple root of nonlinear equations based on Newton-Raphson method. Two modified Newton-Raphson methods for solving nonlinear equations are suggested. Both of the methods are free from second derivatives. Numerical examples are made to show the performance of the presented methods, and to compare with other ones. The numerical results illustrate that the proposed methods are more efficient and performs better than Newton-Raphson method.

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## References

[1] Traub, J. F. 1964. Iterative Methods for Solution of Equations, Prentice-Hall, Englewood Clis, NJ.

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[3] Chun, C. B. 2006. A new iterative method for solving nonlinear equations, Appl. Math. Comput. 178, p. 415-422.

[4] Fang, L., He, G. P., and Hu, Z. Y. 2008. A cubically convergent Newton-type method under weak conditions, J. Comput. Appl. Math. 220, p. 409-412.

[5] Kou, J. S. 2007. The improvements of modified Newton's method, Appl. Math. Comput. 189, p. 602-609.

[6] Abbasbandy, S. 2003. Improving Newton–Raphson method for nonlinear equations by modified Adomian decomposition method, Appl. Math. Comput. 145, p. 887–893.

[7] Fang, L. and He G. P. 2009. Some modifications of Newton's method with higher-order convergence for solving nonlinear equations, J. Comput. Appl. Math. 228, p. 296-303.

[8] Chun, C. B. 2008. A simply constructed third-order modifications of Newton's method, J. Comput. Appl. Math. 219, p. 81-89.

[9] Fang, L., Sun, L. and He G. P. 2008. An efficient Newton-type method with fifth-order for solving nonlinear equations, Comput. Appl. Math. 27, p. 269-274.

[10] Grau, M. and Diaz-Barrero, J.L. 2006. An improvement to Ostrowski root-finding method, Appl. Math. Comput. 173 , p. 450-456.

[2] Potra, F.A., Potra-Pták. 1984. Nondiscrete induction and iterative processes, Research Notes in Mathematics, Vol. 103, Pitman, Boston.

[3] Chun, C. B. 2006. A new iterative method for solving nonlinear equations, Appl. Math. Comput. 178, p. 415-422.

[4] Fang, L., He, G. P., and Hu, Z. Y. 2008. A cubically convergent Newton-type method under weak conditions, J. Comput. Appl. Math. 220, p. 409-412.

[5] Kou, J. S. 2007. The improvements of modified Newton's method, Appl. Math. Comput. 189, p. 602-609.

[6] Abbasbandy, S. 2003. Improving Newton–Raphson method for nonlinear equations by modified Adomian decomposition method, Appl. Math. Comput. 145, p. 887–893.

[7] Fang, L. and He G. P. 2009. Some modifications of Newton's method with higher-order convergence for solving nonlinear equations, J. Comput. Appl. Math. 228, p. 296-303.

[8] Chun, C. B. 2008. A simply constructed third-order modifications of Newton's method, J. Comput. Appl. Math. 219, p. 81-89.

[9] Fang, L., Sun, L. and He G. P. 2008. An efficient Newton-type method with fifth-order for solving nonlinear equations, Comput. Appl. Math. 27, p. 269-274.

[10] Grau, M. and Diaz-Barrero, J.L. 2006. An improvement to Ostrowski root-finding method, Appl. Math. Comput. 173 , p. 450-456.