Newton-type methods on Riemannian manifolds under Kantorovich-type conditions

One of the most studied problems in numerical analysis is the approximation of nonlinear equations using iterative methods. In the last years, attention has been paid in studying Newton's method on manifolds. In this paper, we generalize this study considering some Newton-type iterative methods. A characterization of the convergence under Kantorovich type conditions and optimal estimates of the error are found. Using normal coordinates the order of convergence is derived. The sufficient semilocal convergence criteria are weaker and the majorizing sequences are tighter for the special cases of simplified Newton and Newton methods than in earlier studies such as Argyros (2004, 2007, 2008) [6,8,12] and Kantorovich and Akilov (1964) [32].