- About this Journal ·
- Abstracting and Indexing ·
- Aims and Scope ·
- Annual Issues ·
- Article Processing Charges ·
- Articles in Press ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents
Abstract and Applied Analysis
Volume 2013 (2013), Article ID 682167, 11 pages
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.
- L. Blum, F. Cucker, M. Shub, and S. Smale, Complexity and Real Computation, Springer, New York, NY, USA, 1998.
- 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.
- 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.
- S. Smale, “The fundamental theorem of algebra and complexity theory,” Bulletin of the American Mathematical Society, vol. 4, no. 1, pp. 1–36, 1981.
- 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.
- S. Smale, “Complexity theory and numerical analysis,” Acta Numerica, vol. 6, pp. 523–551, 1997.
- X. Wang, “Convergence on the iteration of Halley family in weak conditions,” Chinese Science Bulletin, vol. 42, no. 7, pp. 552–555, 1997.
- 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.
- V. Candela and A. Marquina, “Recurrence relations for rational cubic methods—I. The Halley method,” Computing, vol. 44, no. 2, pp. 169–184, 1990.
- V. Candela and A. Marquina, “Recurrence relations for rational cubic methods—II. The Chebyshev method,” Computing, vol. 45, no. 4, pp. 355–367, 1990.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.