- 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 367161, 9 pages
Local Convergence of Newton’s Method on Lie Groups and Uniqueness Balls
1Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China
2Department of Mathematics, Zhejiang University of Technology, Hangzhou 310032, China
3Department of Mathematics, National Sun Yat-sen University, Kaohsiung 804, Taiwan
4Center for Fundamental Science, Kaohsiung Medical University, Kaohsiung 80702, Taiwan
5Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia
Received 20 August 2013; Accepted 1 October 2013
Academic Editor: Antonio M. Peralta
Copyright © 2013 Jinsu He 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. V. Kantorovich and G. P. Akilov, Functional Analysis, Pergamon Press, Oxford, UK, 2nd edition, 1982.
- S. Smale, “Newton's method estimates from data at one point,” in The Merging of Disciplines: New Directions in Pure, Applied, and Computational Mathematics (Laramie, Wyo., 1985), R. Ewing, K. Gross, and C. Martin, Eds., pp. 185–196, Springer, New York, NY, USA, 1986.
- J. A. Ezquerro and M. A. Hernández, “Generalized differentiability conditions for Newton's method,” IMA Journal of Numerical Analysis, vol. 22, no. 2, pp. 187–205, 2002.
- J. A. Ezquerro and M. A. Hernández, “On an application of Newton's method to nonlinear operators with -conditioned second derivative,” BIT Numerical Mathematics, vol. 42, no. 3, pp. 519–530, 2002.
- J. M. Gutiérrez and M. A. Hernández, “Newton's method under weak Kantorovich conditions,” IMA Journal of Numerical Analysis, vol. 20, no. 4, pp. 521–532, 2000.
- X. Wang, “Convergence of Newton's method and uniqueness of the solution of equations in Banach space,” IMA Journal of Numerical Analysis, vol. 20, no. 1, pp. 123–134, 2000.
- P. P. Zabrejko and D. F. Nguen, “The majorant method in the theory of Newton-Kantorovich approximations and the Pták error estimates,” Numerical Functional Analysis and Optimization, vol. 9, no. 5-6, pp. 671–684, 1987.
- O. P. Ferreira and B. F. Svaiter, “Kantorovich's theorem on Newton's method in Riemannian manifolds,” Journal of Complexity, vol. 18, no. 1, pp. 304–329, 2002.
- J.-P. Dedieu, P. Priouret, and G. Malajovich, “Newton's method on Riemannian manifolds: convariant alpha theory,” IMA Journal of Numerical Analysis, vol. 23, no. 3, pp. 395–419, 2003.
- C. Li and J. Wang, “Newton's method on Riemannian manifolds: Smale's point estimate theory under the -condition,” IMA Journal of Numerical Analysis, vol. 26, no. 2, pp. 228–251, 2006.
- J. Wang and C. Li, “Uniqueness of the singular points of vector fields on Riemannian manifolds under the -condition,” Journal of Complexity, vol. 22, no. 4, pp. 533–548, 2006.
- C. Li and J. Wang, “Convergence of the Newton method and uniqueness of zeros of vector fields on Riemannian manifolds,” Science in China. Series A, vol. 48, no. 11, pp. 1465–1478, 2005.
- J. H. Wang, “Convergence of Newton's method for sections on Riemannian manifolds,” Journal of Optimization Theory and Applications, vol. 148, no. 1, pp. 125–145, 2011.
- C. Li and J. Wang, “Newton's method for sections on Riemannian manifolds: generalized covariant -theory,” Journal of Complexity, vol. 24, no. 3, pp. 423–451, 2008.
- F. Alvarez, J. Bolte, and J. Munier, “A unifying local convergence result for Newton's method in Riemannian manifolds,” Foundations of Computational Mathematics, vol. 8, no. 2, pp. 197–226, 2008.
- R. E. Mahony, “The constrained Newton method on a Lie group and the symmetric eigenvalue problem,” Linear Algebra and Its Applications, vol. 248, pp. 67–89, 1996.
- B. Owren and B. Welfert, “The Newton iteration on Lie groups,” BIT Numerical Mathematics, vol. 40, no. 1, pp. 121–145, 2000.
- J.-H. Wang and C. Li, “Kantorovich's theorems for Newton's method for mappings and optimization problems on Lie groups,” IMA Journal of Numerical Analysis, vol. 31, no. 1, pp. 322–347, 2011.
- C. Li, J.-H. Wang, and J.-P. Dedieu, “Smale's point estimate theory for Newton's method on Lie groups,” Journal of Complexity, vol. 25, no. 2, pp. 128–151, 2009.
- S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, vol. 80 of Pure and Applied Mathematics, Academic Press, New York, NY, USA, 1978.
- V. S. Varadarajan, Lie Groups, Lie Algebras, and Their Representations, vol. 102 of Graduate Texts in Mathematics, Springer, New York, NY, USA, 1984.
- M. P. do Carmo, Riemannian Geometry, Mathematics: Theory & Applications, Birkhäuser, Boston, Mass, USA, 1992.
- 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.
- J. He, J. H. Wang, and J. C. Yao, “Convergence criteria of Newton's method on Lie groups,” Fixed Point Theory and Applications, to appear.
- X. H. Wang and D. F. Han, “Criterion and Newton's method under weak conditions,” Chinese Journal of Numerical Mathematics and Applications, vol. 19, no. 2, pp. 96–105, 1997.
- X. Wang, “Convergence on the iteration of Halley family in weak conditions,” Chinese Science Bulletin, vol. 42, no. 7, pp. 552–555, 1997.