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Discrete Dynamics in Nature and Society
Volume 2013 (2013), Article ID 301718, 10 pages
http://dx.doi.org/10.1155/2013/301718
Research Article

Some Iterative Methods Free from Derivatives and Their Basins of Attraction for Nonlinear Equations

1Department of Chemistry, Roudehen Branch, Islamic Azad University, Roudehen, Iran
2Department of Mathematics, Zahedan Branch, Islamic Azad University, Zahedan, Iran
3Department of Mathematics and Applied Mathematics, School of Mathematical and Natural Sciences, University of Venda, P. Bag X5050, Thohoyandou 0950, South Africa

Received 4 January 2013; Accepted 2 April 2013

Academic Editor: Fathi Allan

Copyright © 2013 Farahnaz Soleimani 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.

Linked References

  1. A. Cordero, J. L. Hueso, E. Martínez, and J. R. Torregrosa, “A family of derivative-free methods with high order of convergence and its application to nonsmooth equations,” Abstract and Applied Analysis, vol. 2012, Article ID 836901, 15 pages, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. F. Soleymani and S. Shateyi, “Two optimal eighth-order derivative-free classes of iterative methods,” Abstract and Applied Analysis, vol. 2012, Article ID 318165, 14 pages, 2012. View at Zentralblatt MATH · View at MathSciNet
  3. A. Iliev and N. Kyurkchiev, Methods in Numerical Analysis: Selected Topics in Numerical Analysis, LAP LAMBERT Academic Publishing, 2010.
  4. A. T. Tiruneh, W. N. Ndlela, and S. J. Nkambule, “A three point formula for finding roots of equations by the method of least squares,” Journal of Applied Mathematics and Bioinformatics, vol. 2, pp. 213–233, 2012.
  5. B. H. Dayton, T.-Y. Li, and Z. Zeng, “Multiple zeros of nonlinear systems,” Mathematics of Computation, vol. 80, no. 276, pp. 2143–2168, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. J. F. Traub, Iterative Methods for the Solution of Equations, Chelsea Publishing, London, UK, 2nd edition, 1982.
  7. H. T. Kung and J. F. Traub, “Optimal order of one-point and multipoint iteration,” Journal of the Association for Computing Machinery, vol. 21, pp. 643–651, 1974. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. F. Soleymani, S. K. Vanani, and A. Afghani, “A general three-step class of optimal iterations for nonlinear equations,” Mathematical Problems in Engineering, vol. 2011, Article ID 469512, 10 pages, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. F. Soleymani, “Optimized Steffensen-type methods with eighth-order convergence and high efficiency index,” International Journal of Mathematics and Mathematical Sciences, vol. 2012, Article ID 932420, 18 pages, 2012. View at Zentralblatt MATH · View at MathSciNet
  10. P. Jain, “Steffensen type methods for solving non-linear equations,” Applied Mathematics and Computation, vol. 194, no. 2, pp. 527–533, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. S. Wagon, Mathematica in Action, Springer, Berlin, Germany, 3rd edition, 2010.
  12. F. Soleymani, “An efficient twelfth-order iterative method for finding all the solutions of nonlinear equations,” Journal of Computational Methods in Sciences and Engineering, 2012. View at Publisher · View at Google Scholar
  13. S. K. Rahimian, F. Jalali, J. D. Seader, and R. E. White, “A new homotopy for seeking all real roots of a nonlinear equation,” Computers and Chemical Engineering, vol. 35, no. 3, pp. 403–411, 2011. View at Publisher · View at Google Scholar · View at Scopus
  14. A. Cayley, “The Newton-Fourier imaginary problem,” American Journal of Mathematics, vol. 2, article 97, 1879. View at Publisher · View at Google Scholar
  15. M. Trott, The Mathematica Guidebook for Numerics, Springer, New York, NY, USA, 2006. View at MathSciNet
  16. M. L. Sahari and I. Djellit, “Fractal Newton basins,” Discrete Dynamics in Nature and Society, vol. 2006, Article ID 28756, 16 pages, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. J. L. Varona, “Graphic and numerical comparison between iterative methods,” The Mathematical Intelligencer, vol. 24, no. 1, pp. 37–46, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. F. Chicharro, A. Cordero, J. M. Gutiérrez, and J. R. Torregrosa, “Complex dynamics of derivative-free methods for nonlinear equations,” Applied Mathematics and Computation, vol. 219, no. 12, pp. 7023–7035, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  19. J. M. Gutiérrez, M. A. Hernández, and N. Romero, “Dynamics of a new family of iterative processes for quadratic polynomials,” Journal of Computational and Applied Mathematics, vol. 233, no. 10, pp. 2688–2695, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. S. Artidiello, F. Chicharro, A. Cordero, and J. R. Torregrosa, “Local convergence and dynamical analysis of a new family of optimal fourth-order iterative methods,” International Journal of Computer Mathematics, 2013. View at Publisher · View at Google Scholar