Some results on convergence of Newton’s method in Banach spaces are established under the assumption that the derivative of the operators satisfies the radius or center Lipschitz condition with a weak L average.

Article history: Received 14 August 2007 Available online 10 April 2008 Submitted by T.D. Benavides

Let X be a p-divisible group with constant Newton polygon over a normal noetherian scheme S. We prove that there exists an isogeny to X → Y such that Y admits a slope filtration. In case S is regular this was proved by N.Katz for dim S = 1 and by T.Zink for dim S ≥ 1.

Our paper reviews Kallay’s results on a geometric version of the classic Newton-Raphson method, in the context of plane curve queries, e.g. curve-curve intersection, point-curve distance computation. Variants of the geometric Newton-Raphson methods are proposed and empirically verified.

The Hermitian and skew-Hermitian splitting (HSS) method is an unconditionally convergent iteration method for solving large sparse non-Hermitian positive definite system of linear equations. By making use of the HSS iteration as the inner solver for the Newton method, we establish a class of Newton-HSS methods for solving large sparse systems of nonlinear equations with positive definite Jacobi...

Newton-Krylov methods, primarily using the Jacobian-Free Newton-Krylov (JFNK) approximation, are examined as an alternative to the traditional power iteration method for the calculation of the fundamental eigenmode in reactor analysis applications based on diffusion theory. One JFNK approach can be considered an acceleration technique for the standard power iteration as it is “wrapped around” t...

We prove that under semi-local assumptions, the inexact Newton method with a fixed relative residual error tolerance converges Q-linearly to a zero of the non-linear operator under consideration. Using this result we show that Newton method for minimizing a self-concordant function or to find a zero of an analytic function can be implemented with a fixed relative residual error tolerance. In th...

A Ulm-like method is proposed in [13] for solving inverse eigenvalue problems with distinct given eigenvalues. The Ulm-like method avoids solving the Jacobian equations used in Newton-like methods and is shown to be quadratically convergent in the root sense. However, the numerical experiments in [3] only show that the Ulm-like method is comparable to the inexact Newton-like method. In this sho...

The 1669-1670 Newton-Raphson’s method is still used to solve equations systems and unconstrained optimization problems. Since this method, some other algorithms inspired by Newton’s have been proposed: in 1839 Chebyshev developped a high order cubical convergence algorithm, and in 1967 Shamanskii proposed an acceleration of Newton’s method. By considering a Newton-type methods as displacement d...

Full Waveform Inversion (FWI) is a powerful method for reconstructing subsurface parameters from local measurements of the seismic wavefield. This method consists in minimizing a distance between predicted and recorded data. The predicted data is computed as the solution of a wave propagation problem. Conventional numerical methods for the resolution of FWI problems are gradient-based methods, ...