`Journal of Applied MathematicsVolume 2012, Article ID 751975, 15 pageshttp://dx.doi.org/10.1155/2012/751975`
Research Article

## On a New Method for Computing the Numerical Solution of Systems of Nonlinear Equations

1Department of Mathematics, Islamic Azad University, Sirjan Branch, Sirjan, Iran
2Department of Mathematics, Islamic Azad University, Zahedan Branch, Zahedan, Iran
3Department of Mathematics, University of Venda, Private Bag X5050, Thohoyandou 0950, South Africa
4School of Mathematical Sciences, University of KwaZulu-Natal, Private Bag X01, Pietermaritzburg 3209, South Africa

Received 22 June 2012; Revised 17 August 2012; Accepted 27 August 2012

Copyright © 2012 H. Montazeri et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. J. M. Ortega and W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, NY, USA, 1970.
2. S. C. Eisenstat and H. F. Walker, “Globally convergent inexact Newton methods,” SIAM Journal on Optimization, vol. 4, no. 2, pp. 393–422, 1994.
3. M. T. Darvishi and B.-C. Shin, “High-order Newton-Krylov methods to solve systems of nonlinear equations,” Journal of the Korean Society for Industrial and Applied Mathematics, vol. 15, no. 1, pp. 19–30, 2011.
4. Z.-Z. Bai and H.-B. An, “A globally convergent Newton-GMRES method for large sparse systems of nonlinear equations,” Applied Numerical Mathematics, vol. 57, no. 3, pp. 235–252, 2007.
5. F. Toutounian, J. Saberi-Nadjafi, and S. H. Taheri, “A hybrid of the Newton-GMRES and electromagnetic meta-heuristic methods for solving systems of nonlinear equations,” Journal of Mathematical Modelling and Algorithms, vol. 8, no. 4, pp. 425–443, 2009.
6. S. Wagon, Mathematica in Action, Springer, New York, NY, USA, 3rd edition, 2010.
7. H. Binous, “Solution of a system of nonlinear equations using the fixed point method,” 2006, http://library.wolfram.com/infocenter/MathSource/6611/.
8. A. Margaris and K. Goulianas, “Finding all roots of $2×2$ nonlinear algebraic systems using back-propagation neural networks,” Neural Computing and Applications, vol. 21, no. 5, pp. 891–904, 2012.
9. E. Turan and A. Ecder, “Set reduction in nonlinear equations,” . In press, http://arxiv.org/abs/1203.3059v1.
10. B.-C. Shin, M. T. Darvishi, and C.-H. Kim, “A comparison of the Newton-Krylov method with high order Newton-like methods to solve nonlinear systems,” Applied Mathematics and Computation, vol. 217, no. 7, pp. 3190–3198, 2010.
11. D. K. R. Babajee, M. Z. Dauhoo, M. T. Darvishi, and A. Barati, “A note on the local convergence of iterative methods based on Adomian decomposition method and 3-node quadrature rule,” Applied Mathematics and Computation, vol. 200, no. 1, pp. 452–458, 2008.
12. J. R. Sharma, R. K. Guha, and R. Sharma, “An efficient fourth order weighted-Newton method for systems of nonlinear equations,” Numerical Algorithms. In press.
13. F. Soleymani, “Regarding the accuracy of optimal eighth-order methods,” Mathematical and Computer Modelling, vol. 53, no. 5-6, pp. 1351–1357, 2011.
14. A. Cordero, J. L. Hueso, E. Martínez, and J. R. Torregrosa, “A modified Newton-Jarratt's composition,” Numerical Algorithms, vol. 55, no. 1, pp. 87–99, 2010.
15. M. Grau-Sánchez, A. Grau, and M. Noguera, “On the computational efficiency index and some iterative methods for solving systems of nonlinear equations,” Journal of Computational and Applied Mathematics, vol. 236, no. 6, pp. 1259–1266, 2011.
16. J. A. Ezquerro, M. Grau-Sánchez, A. Grau, M. A. Hernández, M. Noguera, and N. Romero, “On iterative methods with accelerated convergence for solving systems of nonlinear equations,” Journal of Optimization Theory and Applications, vol. 151, no. 1, pp. 163–174, 2011.
17. M. Trott, The Mathematica GuideBook for Numerics, Springer, New York, NY, USA, 2006.
18. A. H. Bhrawy, E. Tohidi, and F. Soleymani, “A new Bernoulli matrix method for solving high-order linear and nonlinear Fredholm integro-differential equations with piecewise intervals,” Applied Mathematics and Computation, vol. 219, no. 2, pp. 482–497, 2012.
19. 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, pp. 557–573, 2009.
20. M. Y. Waziri, W. J. Leong, M. A. Hassan, and M. Monsi, “A low memory solver for integral equations of Chandrasekhar type in the radiative transfer problems,” Mathematical Problems in Engineering, vol. 2011, Article ID 467017, 12 pages, 2011.
21. O. R. N. Samadi and E. Tohidi, “The spectral method for solving systems of Volterra integral equations,” Journal of Applied Mathematics and Computing, vol. 40, no. 1-2, pp. 477–497, 2012.