Newton's method for sections on Riemannian manifolds
Newton's method for sections on Riemannian manifolds
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DOI：
10.1016/j.jco.2007.12.003
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发表时间：
2008-06
影响因子：
1.7
通讯作者：
Chong Li;Jinhua Wang
中科院分区：
文献类型：
--
作者：
Chong Li;Jinhua Wang
将一种L-平均Lipschitz条件引入黎曼流形上截面的协变导数。假设截面的协变导数满足这种L-平均Lipschitz条件，建立了牛顿法的收敛判据以及黎曼流形上与曲率无关的截面奇点唯一球半径。提供了一些特殊情况的应用，包括 Kantorovich 条件和 γ 条件以及 Smale 的 α 理论。特别是，Ferreira 和 Svaiter [黎曼流形牛顿法的康托罗维奇定理，J. Complexity 18 (2002) 304–329] 的结果得到了扩展，而 Dedieu Priouret, Malajovich [黎曼流形牛顿法：协变 alpha 理论，IMA J. Numer.] 的结果得到了扩展。肛门。 23 (2003) 395–419] 得到显着改进。此外，由 Alvarez、Bolter、Munier [黎曼流形中牛顿法的统一局部收敛结果，找到了相应的结果。计算。数学。出现]向量场和黎曼流形上的映射也得到了扩展。
One kind of the L-average Lipschitz condition is introduced to covariant derivatives of sections on Riemannian manifolds. A convergence criterion of Newton's method and the radii of the uniqueness balls of the singular points for sections on Riemannian manifolds, which is independent of the curvatures, are established under the assumption that the covariant derivatives of the sections satisfy this kind of the L-average Lipschitz condition. Some applications to special cases including Kantorovich's condition and the γ-condition as well as Smale's α-theory are provided. In particular, the result due to Ferreira and Svaiter [Kantorovich's Theorem on Newton's method in Riemannian manifolds, J. Complexity 18 (2002) 304–329] is extended while the results due to Dedieu Priouret, Malajovich [Newton's method on Riemannian manifolds: covariant alpha theory, IMA J. Numer. Anal. 23 (2003) 395–419] are improved significantly. Moreover, the corresponding results due to Alvarez, Bolter, Munier [A unifying local convergence result for Newton's method in Riemannian manifolds, Found. Comput. Math. to appear] for vector fields and mappings on Riemannian manifolds are also extended.