Newton's Method On Manifolds: Generalized Point Estimate Theory

The local and semi-local convergence criterions of Newton's method on Riemannian man-ifolds and Lie groups are given, and the generalized point estimate theories of Newton'smethod on Riemannian manifolds and Lie groups are provided. The main work done in thisdissertation is organized as follows:In Chapter 1, we shall study Newton's method for sections on Riemannian manifoldswhich clearly includes vector fields and vector valued maps on Riemannian manifolds asspecial cases. The main tool is the notion of the Lipschitz condition with positive non-decreasing integrable function L(·)-average along the piecewise geodesic which includes theclassical Lipschitz condition as well as theγ-condition as special cases. Under the assumptionthat the covariant derivative of the sections satisfies the Lipschitz condition with L-averagealong the piecewise geodesic, a unified convergence criterion for Newton's method and theradii of the uniqueness balls of singular points around the initial points of sections are es-tablished. Some applications to special cases, which include the Kantorovich's condition andtheγ-condition as well as the Smale'sα-theory for sections on Riemannian manifolds, areprovided. Furthermore, under the assumption that the covariant derivative of the sectionssatisfies the Lipschitz condition with L-average along the geodesic, the estimates of the radiiof convergence balls of Newton's method and uniqueness balls of singular points around thesingular points of sections on Riemannian manifolds are given. Some classical results suchas the Kantorovich's type theorems and the theorems under theγcondition as well as theSmale'sγ-theory are extended.In Chapter 2, we will use one-parameter subgroups of the Lie group to develop a version ofNewton's method on an arbitrary Lie group for maps from a Lie group to its correspondingLie algebra which is independent of affine connections on the Lie group and differs fromNewton's method for sections on Riemannian manifolds. The main tool is the notion of theLipschitz condition with positive nondecreasing integrable function L(·)-average along thepieces one-parameter subgroup which includes the classical Lipschitz condition as well as theγ-condition as special cases. Under the assumption that the differential of the maps satisfies the Lipschitz condition with L-average along the pieces one-parameter subgroup, a unifiedconvergence criterion for Newton's method is established. Some applications to special cases,which include the Kantorovich's condition and theγ-condition as well as the Smale'sα-theoryfor maps from a Lie group to its corresponding Lie algebra, are provided. Furthermore,under the assumption that the differential of the maps satisfies the Lipschitz condition withL-average along the one-parameter subgroup, the estimates of the radii of convergence ballsof Newton's method are given. Some classical results such as the Kantorovich's type theoremand the theorems under theγ-condition as well as the Smale'sγ-theory are extended.