Uniqueness of the singular points of vector fields on Riemannian manifolds under the γ-condition

Journal of Complexity

Volumn 22, Issue 4, 2006, Pages 533-548

  Wang, Jin hua ^a,b   Li, Chong ^a  

^a ZHEJIANG UNIVERSITY   (China)

^b ZHEJIANG UNIVERSITY OF TECHNOLOGY   (China)

Author keywords

Newton's method; Riemannian mainfold; The comdition; Uniqueness of the singular point; Vector field

Indexed keywords

NUMERICAL METHODS; PROBLEM SOLVING; VECTORS;

NEWTON'S METHOD; RIEMANNIAN MAINFOLD; UNIQUENESS OF THE SINGULAR POINT; VECTOR FIELD;

COMPUTATIONAL COMPLEXITY;

EID: 33746022694     PISSN: 0885064X     EISSN: 10902708     Source Type: Journal    
DOI: 10.1016/j.jco.2005.11.004     Document Type: Article

References (19)

1. Blum L., Cucker F., Shub M., and Smale S. Complexity and Real Computation (1997), Springer, New York
2. Boothby W.M. An Introduction to Differentiable Manifolds and Riemannian Geometry. second ed. (1986), Academic Press, New York
3. Dedieu J.P., Priouret P., and Malajovich G. Newton's method on Riemannian manifolds: covariant alpha theory. IMA J Numer Anal. 23 (2003) 395-419
4. DoCarmo M.P. Riemannian Geometry (1992), Birkhauser, Boston
5. Edelman A., Arias T.A., and Smith T. The geometry of algorithms with orthogonality constraints. SIAM J. Matrix Anal. Appl. 20 2 (1998) 303-353
6. Ferreira O.P., and Svaiter B.F. Kantorovich's theorem on Newton's method in Riemannian manifolds. J. Complex. 18 (2002) 304-329
7. Gabay D. Minimizing a differentiable function over a differential manifold. J. Optim. Theory Appl. 37 2 (1982) 177-219
8. Kantorovich L.V., and Akilov G.P. Functional Analysis (1982), Pergamon, Oxford
9. Lang S. Differential and Riemannian Manifolds, GTM 160 (1995), Springer, New York
10. C. Li, J.H. Wang, Newton's method on Riemannian manifolds: Smale's point estimate theory under the γ-condition, IMA J. Number Anal., 26(2) (2006) 228-251.
11. Smale S. The fundamental theorem of algebra and complexity theory. Bull. AMS. 4 (1981) 1-36
12. Smale S. Newton's method estimates from data at one point. In: Ewing R., Gross K., and Martin C. (Eds). The Merging of Disciplines: New Directions in Pure, Applied and Computational Mathematics (1986), Springer, New York 185-196
13. Smale S. Complexity theory and numerical analysis. Acta Numer. 6 (1997) 523-551
14. S.T. Smith, Geometric optimization method for adaptive filtering, Ph.D. Thesis, Harvard University, Cambridge, MA, 1993.
15. S.T. Smith, Optimization techniques on Riemannian Manifolds, in: Anthony Bloch (ed.), Fields Institute Communications, vol. 3, pp. 113-146, American Mathematical Society, Providence, RI, 1994.
16. C. Udriste, Convex functions and optimization methods on Riemannian manifolds, Mathematics and its Applications, vol. 297, Kluwer Academic, Dordrecht, 1994.
17. Wang X.H. Convergence of Newton's method and inverse function theorem in Banach space. Math. Comput. 225 68 (1999) 169-186
18. Wang X.H., and Han D.F. On the dominating sequence method in the point estimates and Smale's theorem. Scientia Sinica Ser. A 33 2 (1990) 135-144
19. Wang X.H., and Han D.F. Criterion α and Newton's method under weak conditions. Chinese J. Numer. Appl. Math. 19 2 (1997) 96-105