Convergence of Newton’s Method for Sections on Riemannian Manifolds
Convergence of Newton’s Method for Sections on Riemannian Manifolds
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DOI：
10.1007/s10957-010-9748-4
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发表时间：
2011
影响因子：
1.9
通讯作者：
Jinhua Wang
中科院分区：
文献类型：
--
作者：
Jinhua Wang
本文研究了黎曼流形上牛顿法的收敛性问题和截面奇点的唯一性问题。假设截面的协变导数满足广义Lipschitz条件。对牛顿法的收敛球和奇点的唯一性球进行了估计。给出了Kantorovich条件、γ-条件以及Smale γ-理论在黎曼流形上的一些特殊应用。特别地，这里的估计完全独立于基础黎曼流形的截面曲率，并且由于Dedieu，Priouret和Malajovich（IMA J. Numer. Anal. 23：395-419，2003），以及Li和Wang（Sci.中国A. 48（11）：1465-1478，2005）。
The present paper is concerned with the convergence problems of Newton’s method and the uniqueness problems of singular points for sections on Riemannian manifolds. Suppose that the covariant derivative of the sections satisfies the generalized Lipschitz condition. The convergence balls of Newton’s method and the uniqueness balls of singular points are estimated. Some applications to special cases, which include the Kantorovich’s condition and theγ-condition, as well as the Smale’sγ-theory for sections on Riemannian manifolds, are given. In particular, the estimates here are completely independent of the sectional curvature of the underlying Riemannian manifold and improve significantly the corresponding ones due to Dedieu, Priouret and Malajovich (IMA J. Numer. Anal. 23:395–419, 2003), as well as the ones in Li and Wang (Sci. China Ser. A. 48(11):1465–1478, 2005).