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IMA Journal of Numerical Analysis Advance Access first published online on October 11, 2009
This version published online on November 16, 2009
IMA Journal of Numerical Analysis, doi:10.1093/imanum/drp015
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© The author 2009. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

Kantorovich's theorems for Newton's method for mappings and optimization problems on Lie groups

Jin-Hua Wang{dagger}

Department of Mathematics, Zhejiang University of Technology, Hangzhou 310032, People's Republic of China

Chong Li{ddagger}

Department of Mathematics, Zhejiang University, Hangzhou 310027, People's Republic of China

{dagger} Corresponding author. Email: wjh{at}zjut.edu.cn

{ddagger} Email: cli{at}zju.edu.cn

Received on 4 June 2008. Revised on 1 April 2009.


  Abstract

With the classical assumptions on f, a convergence criterion of Newton's method (independent of affine connections) to find zeros of a mapping f from a Lie group to its Lie algebra is established, and estimates of the convergence domains of Newton's method are obtained, which improve the corresponding results in Owren & Welfert (2000, BIT Numer. Math., 40, 121–145) and Wang & Li (2006, J. Zhejiang Univ. Sci. A, 8, 978–986). Applications to optimization are provided and the results due to Mahony (1996, Linear Algebra Appl., 248, 67–89) are extended and improved accordingly.

Key Words: Newton's method; Lie group; Lipschitz condition

The original title of the paper has been updated in the new version.


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