Convergence of Newton’s Method for Sections on Riemannian Manifolds

J. H. Wang ()
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J. H. Wang: Zhejiang University of Technology

Journal of Optimization Theory and Applications, 2011, vol. 148, issue 1, No 8, 125-145

Abstract: Abstract 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 the γ-condition, as well as the Smale’s γ-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).

Keywords: Newton’s method; Riemannian manifold; Section; Generalized Lipschitz condition (search for similar items in EconPapers)
Date: 2011
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Citations: View citations in EconPapers (7)

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DOI: 10.1007/s10957-010-9748-4

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