Newton's method on Riemannian manifolds: Smale's point estimate theory under the γ-condition

IMA Journal of Numerical Analysis

Volumn 26, Issue 2, 2006, Pages 228-251

  Li, Chong ^a   Wang, Jinhua ^a  

^a ZHEJIANG UNIVERSITY   (China)

Author keywords

Newton's method; Riemannian manifold; Smale's point estimate theory; The condition; Vector field

Indexed keywords

NEWTON-RAPHSON METHOD;

ALPHA THEORY; NEWTON'S METHODS; POINT ESTIMATE; RIEMANNIAN MANIFOLD; VECTOR FIELDS;

GEOMETRY;

EID: 33645732823     PISSN: 02724979     EISSN: 14643642     Source Type: Journal    
DOI: 10.1002/adma.200500202     Document Type: Article

References (18)

1. BLUM, L., CUCKER, F., SHUB, M. & SMALE, S. (1997) Complexity and Real Computation. New York: Springer.
2. BOOTHBY, W. M. (1986) An Introduction to Differentiable Manifolds and Riemannian Geometry, 2nd edn. New York: Academic Press, Inc.
3. DEDIEU, J. P., PRIOURET, P. & MALAJOVICH, G. (2003) Newton's method on Riemannian manifolds: covariant alpha theory. IMA J. Numer. Anal., 23, 395-419.
4. DOCARMO, M. P. (1992) Riemannian Geometry. Boston, MA: Birkhauser.
5. EDELMAN, A., ARIAS, T. A. & SMITH, T. (1998) The geometry of algorithms with orthogonality constraints. SIAM J. Matrix Anal. Appl., 20, 303-353.
6. FERREIRA, O. P. & SVAITER, B. F. (2002) Kantorovich's theorem on Newton's method in Riemannian manifolds. J. Complex., 18, 304-329.
7. GABAY, D. (1982) Minimizing a differentiable function over a differential manifold. J. Optim. Theory Appl., 37, 177-219.
8. KANTOROVICH, L. V. & AKILOV, G. P. (1982) Functional Analysis. Oxford: Pergamon.
9. LANG, S. (1995) Differential and Riemannian Manifolds, GTM 160. New York: Springer.
10. SMALE, S. (1981) The fundamental theorem of algebra and complexity theory. Bull. Am. Math. Soc., 4, 1-36.
11. SMALE, S. (1986) Newton's method estimates from data at one point. The Merging of Disciplines: New Directions in Pure, Applied and Computational Mathematics (R. Ewing, K. Gross & C. Martin eds). New York: Springer, pp. 185-196.
12. SMALE, S. (1997) Complexity theory and numerical analysis. Acta Numer., 6, 523-551.
13. SMITH, S. T. (1993) Geometric optimization method for adaptive filtering. Ph.D. Thesis, Harvard University, Cambridge, MA.
14. SMITH, S. T. (1994) Optimization techniques on Riemannian manifolds. Fields Institute Communications, vol. 3. (A. Bloch ed.) Providence, RI: American Mathematical Society, pp. 113-146.
15. UDRISTE, C. (1994) Convex Functions and Optimization Methods on Riemannian Manifolds. Dordrecht: Kluwer Academic.
16. WANG, X. H. (1999) Convergence of Newton's method and inverse function theorem in Banach space. Math. Comput., 225, 169-186.
17. WANG, X. H. & HAN, D. F. (1990) On the dominating sequence method in the point estimates and Smale's theorem. Sci. Sin. Ser. A, 33, 135-144.
18. WANG, X. H. & HAN, D. F. (1997) Criterion α and Newton's method. Chin. J. Numer. Appl. Math., 19, 96-105.