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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 682167, 11 pages
http://dx.doi.org/10.1155/2013/682167
Research Article

Convergence Behavior for Newton-Steffensen’s Method under -Condition of Second Derivative

1Department of Mathematics, Zhejiang University, Hangzhou 310027, China
2Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China

Received 2 September 2013; Accepted 30 September 2013

Academic Editor: Jen-Chih Yao

Copyright © 2013 Yonghui Ling 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. L. Blum, F. Cucker, M. Shub, and S. Smale, Complexity and Real Computation, Springer, New York, NY, USA, 1998. View at MathSciNet
  2. J.-S. He, J.-H. Wang, and C. Li, “Newton's method for underdetermined systems of equations under the γ-condition,” Numerical Functional Analysis and Optimization, vol. 28, no. 5-6, pp. 663–679, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  3. C. Li, N. Hu, and J. Wang, “Convergence behavior of Gauss-Newton's method and extensions of the Smale point estimate theory,” Journal of Complexity, vol. 26, no. 3, pp. 268–295, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. S. Smale, “The fundamental theorem of algebra and complexity theory,” Bulletin of the American Mathematical Society, vol. 4, no. 1, pp. 1–36, 1981. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. S. Smale, “Newton's method estimates from data at one point,” in The Merging of Disciplines: New Directions in Pure, Applied, and Computational Mathematics, pp. 185–196, Springer, New York, NY, USA, 1986. View at Zentralblatt MATH · View at MathSciNet
  6. S. Smale, “Complexity theory and numerical analysis,” Acta Numerica, vol. 6, pp. 523–551, 1997.
  7. X. Wang, “Convergence on the iteration of Halley family in weak conditions,” Chinese Science Bulletin, vol. 42, no. 7, pp. 552–555, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. X. H. Wang and D. F. Han, “Criterion α and Newton's method under weak conditions,” Mathematica Numerica Sinica, vol. 19, no. 1, pp. 96–105, 1997.
  9. V. Candela and A. Marquina, “Recurrence relations for rational cubic methods—I. The Halley method,” Computing, vol. 44, no. 2, pp. 169–184, 1990. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. V. Candela and A. Marquina, “Recurrence relations for rational cubic methods—II. The Chebyshev method,” Computing, vol. 45, no. 4, pp. 355–367, 1990. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. M. A. Hernández and M. A. Salanova, “A family of Chebyshev-Halley type methods,” International Journal of Computer Mathematics, vol. 47, pp. 59–63, 1993.
  12. H. Wang, C. Li, and X. Wang, “On relationship between convergence ball of Euler iteration in Banach spaces and its dynamical behavior on Riemann spheres,” Science in China. Mathematics, vol. 46, no. 3, pp. 376–382, 2003. View at Zentralblatt MATH · View at MathSciNet
  13. X. Wang and C. Li, “On the united theory of the family of Euler-Halley type methods with cubical convergence in Banach spaces,” Journal of Computational Mathematics, vol. 21, no. 2, pp. 195–200, 2003. View at Zentralblatt MATH · View at MathSciNet
  14. X. Xu and Y. Ling, “Semilocal convergence for Halley's method under weak Lipschitz condition,” Applied Mathematics and Computation, vol. 215, no. 8, pp. 3057–3067, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. X. Xu and Y. Ling, “Semilocal convergence for a family of Chebyshev-Halley like iterations under a mild differentiability condition,” Journal of Applied Mathematics and Computing, vol. 40, no. 1-2, pp. 627–647, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  16. X. Ye and C. Li, “Convergence of the family of the deformed Euler-Halley iterations under the Hölder condition of the second derivative,” Journal of Computational and Applied Mathematics, vol. 194, no. 2, pp. 294–308, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. J. R. Sharma, “A composite third order Newton-Steffensen method for solving nonlinear equations,” Applied Mathematics and Computation, vol. 169, no. 1, pp. 242–246, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. X. H. Wang and D. F. Han, “On dominating sequence method in the point estimate and Smale theorem,” Science in China. Mathematics, vol. 33, no. 2, pp. 135–144, 1990. View at Zentralblatt MATH · View at MathSciNet
  19. X. Wang, “Convergence of Newton's method and inverse function theorem in Banach space,” Mathematics of Computation, vol. 68, no. 225, pp. 169–186, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. X. H. Wang, D. F. Han, and F. Y. Sun, “Point estimates for some deformation Newton iterations,” Mathematica Numerica Sinica, vol. 12, no. 2, pp. 145–156, 1990. View at MathSciNet