Convergence of the Newton method and uniqueness of zeros of vector fields on Riemannian manifolds

Science in China, Series A: Mathematics

Volumn 48, Issue 11, 2005, Pages 1465-1478

  Li, Chong ^a   Wang, Jinhua ^a  

^a ZHEJIANG UNIVERSITY   (China)

Author keywords

Convergence ball; Newton method; Riemannian manifold; Uniqueness ball

Indexed keywords

EID: 29144500438     PISSN: 10069283     EISSN: None     Source Type: Journal    
DOI: 10.1360/04ys0147     Document Type: Article

References (20)

1. Kantorovich, L. V., Akilov, G. P., Functional Analysis, Oxford: Pergamon, 1982.
2. Smale, S., Newton's method estimates from data at one point, in The Merging of Disciplines: New Directions in Pure, Applied and Computational Mathematics (eds. Ewing, R. Gross, K. Martin, C.), New York: Springer, 1986, 185-196.
3. Traub, J. F., Wozniakowski, H., Convergence and complexity of Newton iteration, J. Assoc. Comput. Math., 1979, 29: 250-258.
4. Wang, X. H., The convergence on Newton's method, Kexue Tongbao (A Special Issue of Mathematics, Physics and Chemistry, in Chinese), 1980, 25: 36-37.
5. Wang, X. H., Convergence of Newton's method and uniqueness of the solution of equations in Banach space, IMA J. Numer Anal., 2000, 20(1): 123-134.
6. Wang, X. H., Convergence of Newton's method and inverse function theorem in Banach space, Mathematics of Computation., 1999, 225(68): 169-186.
7. Edelman, A., Arias, T. A., Smith, T., The geometry of algorithms with orthogonality constraints, SIAM J. Matrix Anal. Appl., 1998, 20(2): 303-353.
8. Gabay, D., Minimizing a differentiable function over a differential manifold, J. Optim. Theory Appl., 1982, 37(2): 177-219.
9. Smith, S. T., Optimization techniques on Riemannian manifolds, Fields institute communications 3, AMS., 1994, 113-146.
10. Smith, S. T., Geometric optimization method for adaptive filtering, Ph. D. thesis, Harvard University, Cambridge, MA, 1993.
11. Udriste, C., Convex functions and optimization methods on Riemannian manifolds, in Mathematics and Its Applications, Vol. 297, Dordrecht: Kluwer Academic, 1994.
12. Ferreira, O. P., Svaiter, B. F., Kantorovich's theorem on Newton's method in Riemannian manifolds, Journal of Complexity., 2002, 18: 304-329.
13. Dedieu, J. P., Priouret, P., Malajovich, G., Newton's method on Riemannian manifolds: Covariant alpha theory, IMA Journal of Numerical Analysis., 2003, 23: 395-419.
14. Chen, W. H., Li, X. X., An Introduction to Riemannian Geometry(in Chinese), Beijing: Beijing University Press, 2002.
15. DoCarmo, M. P., Riemannian Geometry, Boston: Birkhauser, 1992.
16. Smale, S., The fundamental theorem of algebra and complexity theory, Bull. AMS, 1981, 4: 1-36.
17. Wang, X. H., Xuan, X. H., Random polynomial space and computational complexity theory, Science in China, Ser. A, 1987, 30: 673-684.
18. Wang, X. H., Li, C., Local and global behaviors for algorithms of solving equations, Chinese Science Bulletin, 2001, 46(6): 441-448.
19. Smale, S., Complexity theory and numerical analysis, Acta Numer., 1997, 6: 523-551.
20. Blum, L., Cucker, F., Shub, M., Smale, S., Complexity and Real Computation, New York: Springer-Verlag, 1997.