Geometric optimization methods for adaptive filtering
Geometric optimization methods for adaptive filtering
Approximate zeros of quadratically convergent algorithms
Mathematics of Computation
A new semilocal convergence theorem for Newton's method
Journal of Computational and Applied Mathematics
Complexity and real computation
Complexity and real computation
The Geometry of Algorithms with Orthogonality Constraints
SIAM Journal on Matrix Analysis and Applications
Convergence of Newton's method and inverse function theorem in Banach space
Mathematics of Computation
Kantorovich's theorem on Newton's method in Riemannian Manifolds
Journal of Complexity
A Unifying Local Convergence Result for Newton's Method in Riemannian Manifolds
Foundations of Computational Mathematics
Gauss---Newton method for convex composite optimizations on Riemannian manifolds
Journal of Global Optimization
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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 @c-condition as well as Smale's @a-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.