The majorant method in the theory of newton-kantorovich approximations and the pták error estimates

A Unifying Local Convergence Result for Newton's Method in Riemannian Manifolds

“…where q is given by (13); see, for instance, [10,16,23,25]. Therefore, Theorem 3.1(vi) is a direct consequence of the following result.…”

“…Condition (5) is a scaled Riemannian analogue of the property used by Zabrejko and Nguen in [25] (see also [2,24]) to prove a local convergence result for Newton's method in Banach spaces, based on a radial parametrization of the original "majorant method" developed by Kantorovich [12]. Here, scaling means that the inverse of X (p 0 ) is incorporated in the distance between covariant derivatives of X, an idea that has been already used in the Banach space context (see [4,23]).…”

“…Under (8) it is elementary to establish the following result (see, for instance, [25,Proposition 3] or Lemma 4.1(i)-(ii) below).…”

“…The striking feature of the majorant method is that the analysis of (3) and (N ) reduces to the analysis of an appropriate scalar equation together with the corresponding scalar Newton iteration. Following the idea of [25], we set…”

“…This technique was originally developed by Zabrejko and Nguen in [25] (see also [2,24]) in order to obtain refinements of the Kantorovich theorem in Banach spaces. In fact, this approach is based on the construction of a real-valued function, namely the majorant function, using an adequate local Lipschitz-type radial estimate for the first derivative of X.…”