Journal of Zhejiang University SCIENCE A
ISSN 1673-565X(Print), 1862-1775(Online), Monthly

2007   Vol. 8   No. 6   p. 978~986

Kantorovich’s theorem for Newton’s method on Lie groups

WANG Jin-hua¹, LI Chong²

(¹Department of Mathematics, Zhejiang University of Technology, Hangzhou 310032, China)
(²Department of Mathematics, Zhejiang University, Hangzhou 310027, China)
E-mail: wjh@zjut.edu.cn; cli@zju.edu.cn
Received Sept. 20, 2006 revision accepted Jan. 4, 2007

Abstract: The convergence criterion of Newton’s method to find the zeros of a map f from a Lie group to its corresponding Lie algebra is established under the assumption that f satisfies the classical Lipschitz condition, and that the radius of convergence ball is also obtained. Furthermore, the radii of the uniqueness balls of the zeros of f are estimated. Owren and Welfert (2000) stated that if the initial point is close sufficiently to a zero of f, then Newton’s method on Lie group converges to the zero; while this paper provides a Kantorovich’s criterion for the convergence of Newton’s method, not requiring the existence of a zero as a priori.

Key words: Newton’s method, Lie group, Kantorovich’s theorem, Lipschitz condition
doi:10.1631/jzus.2007.A0978             CLC number: O242.23

References:

[1] Adler, R.L., Dedieu, J.P., Margulies, J.Y., Martens, M., Shub, M., 2002. Newton’s method on Riemannian manifolds and a geometric model for the human spine. IMA J. Numer. Anal., 22(3):359-390.

[2] Dedieu, J.P., Priouret, P., Malajovich, G., 2003. Newton’s method on Riemannian manifolds: covariant alpha theory. IMA J. Numer. Anal., 23:395-419.

[3] Edelman, A., Arias, T.A., Smith, T., 1998. The geometry of algorithms with orthogonality constraints. SIAM J. Matrix Anal. Appl., 20:303-353.

[4] Ferreira, O.P., Svaiter, B.F., 2002. Kantorovich’s theorem on Newton’s method in Riemannian manifolds. J. Complexity, 18:304-329.

[5] Gabay, D., 1982. Minimizing a differentiable function over a differential manifold. J. Optim. Theory Appl., 37:177-219.

[6] Gutierrez, J.M., Hernandez, M.A., 2000. Newton’s method under weak Kantorovich conditions. IMA J. Numer. Anal., 20:521-532.

[7] Kantorovich, L.V., 1948. On Newton method for functional equations. Dokl. Acad. Nauk, 59(7):1237-1240.

[8] Kantorovich, L.V., Akilov, G.P., 1982. Functional Analysis. Pergamon, Oxford.

[9] Li, C., Wang, J.H., 2005. Convergence of Newton’s method and uniqueness of zeros of vector fields on Riemannian manifolds. Sci. in China (Ser. A), 48(11):1465-1478.

[10] Li, C., Wang, J.H., 2006. Newton’s method on Riemannian manifolds: Smale’s point estimate theory under the γ-condition. IMA J. Numer. Anal., 26(2):228-251.

[11] Mahony, R.E., 1996. The constrained Newton method on a Lie group and the symmetric eigenvalue problem. Linear Algebra Appl., 248:67-89.

[12] Owren, B., Welfert, B., 2000. The Newton iteration on Lie groups. BIT Numer. Math., 40:121-145.

[13] Smale, S., 1986. Newton’s Method Estimates from Data at One Point. In: Ewing, R., Gross, K., Martin, C. (Eds.), The Merging of Disciplines: New Directions in Pure, Applied and Computational Mathematics. Springer, New York, p.185-196.

[14] Smith, S.T., 1993. Geometric Optimization Method for Adaptive Filtering. Ph.D Thesis, Harvard University, Cambridge, MA.

[15] Smith, S.T., 1994. Optimization Techniques on Riemannian Manifolds. Fields Institute Communications. American Mathematical Society, Providence, RI, 3:113-146.

[16] Udriste, C., 1994. Convex Functions and Optimization Methods on Riemannian Manifolds. Mathematics and its Applications, Vol. 297, Kluwer Academic, Dordrecht.

[17] Varadarajan, V.S., 1984. Lie Groups, Lie Algebras and Their Representations. GTM No. 102. Springer-Verlag, New York.

[18] Wang, X.H., 1997. Convergence on the iteration of Halley family in weak conditions. Chin. Sci. Bull., 42:552-555. (in Chinese).

[19] Wang, X.H., Han, D.F., 1997. Criterion α and Newton’s method in weak condition. Chin. J. Numer. Appl. Math., 19:96-105.

[20] Wang, X.H., 1999. Convergence of Newton’s method and inverse function theorem in Banach space. Math. Comput., 68:169-186.

[21] Wang, J.H., Li, C., 2006. Uniqueness of the singular point of vector field on Riemannian manifold under the γ-condition. J. Complexity, 22:533-548.

[22] Warner, F.W., 1983. Foundations of Differentiable Manifolds and Lie Groups. GTM No. 94. Springer-Verlag, New York.