Smale's point estimate theory for Newton's method on Lie groups

Journal of Complexity

Volumn 25, Issue 2, 2009, Pages 128-151

  Li, Chong ^a   Wang, Jin Hua ^b   Dedieu, Jean Pierre ^c  

^a ZHEJIANG UNIVERSITY   (China)

^b ZHEJIANG UNIVERSITY OF TECHNOLOGY   (China)

^c INSTITUT DE MATHÉMATIQUES DE TOULOUSE   (France)

Author keywords

Lie group; Newton's method; Smale's point estimate theory; The condition

Indexed keywords

INITIAL VALUE PROBLEMS; NEWTON-RAPHSON METHOD;

ALPHA THEORY; LIE ALGEBRA; NEWTON'S METHODS; POINT ESTIMATE;

LIE GROUPS;

EID: 62549157073     PISSN: 0885064X     EISSN: 10902708     Source Type: Journal    
DOI: 10.1016/j.jco.2008.11.001     Document Type: Article

References (26)

1. Adler R., Dedieu J.P., Margulies J., Martens M., and Shub M. Newton method on Riemannian manifolds and a geometric model for human spine. IMA J. Numer. Anal. 22 (2002) 1-32
2. Edelman A., Arias T.A., and Smith T. The geometry of algorithms with orthogonality constraints. SIAM J. Matrix Anal. Appl. 20 (1998) 303-353
3. Gabay D. Minimizing a differentiable function over a differential manifold. J. Optim. Theory Appl. 37 (1982) 177-219
4. Smith S.T. Optimization techniques on Riemannian manifolds. Fields Institute Communications vol. 3 (1994), American Mathematical Society, Providence, RI 113-146
5. S.T. Smith, Geometric optimization method for adaptive filtering, Ph. D. Thesis, Harvard University, Cambridge, MA, (1993)
6. Udriste C. Convex Functions and Optimization Methods on Riemannian Manifolds. Mathematics and Its Applications vol. 297 (1994), Kluwer Academic, Dordrecht
7. Kantorovich L.V. On Newton method for functional equations. Dokl. Acad. Nauk. 59 (1948) 1237-1240
8. Kantorovich L.V., and Akilov G.P. Functional Analysis (1982), Pergamon, Oxford
9. Ferreira O.P., and Svaiter B.F. Kantorovich's theorem on Newton's method in Riemannian manifolds. J. Complexity. 18 (2002) 304-329
10. Blum L., Cucker F., Shub M., and Smale S. Complexity and Real Computation (1997), Springer-Verlag, New York
11. 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
12. 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
13. Li C., and Wang J.H. Newton's method on Riemannian manifolds: Smale's point estimate theory under the γ-condition. IMA J. Numer. Anal. 26 (2006) 228-251
14. Wang X.H., and Han D.F. Criterion α and Newton's method in weak condition. Chinese J. Numer. Appl. Math. 19 (1997) 96-105
15. Alvarez F., Bolte J., and Munier J. A unifying local convergence result for Newton's method in Riemannian manifolds. Found. Comput. Math. 8 (2008) 197-226
16. Mahony R.E. The constrained Newton method on a Lie group and the symmetric eigenvalue problem. Linear Algebra Appl. 248 (1996) 67-89
17. Mahony R.E., and Manton J. The geometry of the Newton method on non-compact Lie groups. J. Global Optim. 23 (2002) 309-327
18. Munthe-Kaas H. High order Runge-Kutta methods on manifold. Appl. Numer. Math. 29 (1999) 115-127
19. Owren B., and Welfert B. The Newton iteration on Lie groups. BIT, Numer. 40 (2000) 121-145
20. Varadarajan V.S. Lie Groups Lie Algebras and their Representations. GTM 102 (1984), Springer-Verlag, New York
21. Warner F.W. Foundations of Differentiable Manifolds and Lie Groups. GTM vol. 94 (1983), Springer-Verlag, New York
22. DoCarmo M.P. Riemannian Geometry (1992), Birkhauser, Boston
23. Wang X.H. Convergence on the iteration of Halley family in weak conditions. Chinese Sci. Bull. 42 (1997) 552-555
24. 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 (1990) 135-144
25. Helgason S. Differential Geometry Lie Groups Symmetric Spaces (1978), Academic Press Inc., New York
26. Hall B.C. Lie Groups Lie Algebras Representations. GTM vol. 222 (2004), Springer-Verlag, New York