Kantorovich’s theorem on Newton’s method under majorant condition in Riemannian manifolds

Extension of concepts and techniques of linear spaces for the Riemannian setting has been frequently attempted. One reason for the extension of such techniques is the possibility to transform some Euclidean non-convex or quasi-convex problems into Riemannian convex problems. In this paper, a version of Kantorovich’s theorem on Newton’s method for finding a singularity of differentiable vector fields defined on a complete Riemannian manifold is presented. In the presented analysis, the classical Lipschitz condition is relaxed using a general majorant function, which enables us to not only establish the existence and uniqueness of the solution but also unify earlier results related to Newton’s method. Moreover, a ball is prescribed around the points satisfying Kantorovich’s assumptions and convergence of the method is ensured for any starting point within this ball. In addition, some bounds for the Q-quadratic convergence of the method, which depends on the majorant function, are obtained.     

On the local convergence of Newton’s method under generalized conditions of Kantorovich

Following an idea similar to that given by Dennis and Schnabel (1996) in [2], we prove a local convergence result for Newton’s method under generalized conditions of Kantorovich type.     

Newton’s method on Lie groups

The purpose of this paper is to study the strong convergence of fixed points for a family of demi-continuous pseudo-contractions by hybrid projection algorithms in the framework of Hilbert spaces. Our results improve and extend the corresponding results announced by many others.     

A Newton’s method for the continuous quadratic knapsack problem

We introduce a new efficient method to solve the continuous quadratic knapsack problem. This is a highly structured quadratic program that appears in different contexts. The method converges after $O(n)$ iterations with overall arithmetic complexity $O(n^2)$ . Numerical experiments show that in practice the method converges in a small number of iterations with overall linear complexity, and is faster than the state-of-the-art algorithms based on median finding, variable fixing, and secant techniques.     

Local convergence of Newton’s method for subanalytic variational inclusions

This paper concerns variational inclusions of the form where f is a single locally Lipschitz subanalytic function and F is a set-valued map acting in Banach spaces. We prove the existence and the convergence of a sequence (x _k ) satisfying where lies to which is the Clarke Jacobian of f at the point x _k .      

Harnack’s principle for quasiminimizers

Abstract We study Harnack type properties of quasiminimizers of the -Dirichlet integral on metric measure spaces equipped with a doubling measure and supporting a Poincaré inequality. We show that an increasing sequence of quasiminimizers converges locally uniformly to a quasiminimizer, provided the limit function is finite at some point, even if the quasiminimizing constant and the boundary values are allowed to vary in a bounded way. If the quasiminimizing constants converge to one, then the limit function is the unique minimizer of the -Dirichlet integral. In the Euclidean case with the Lebesgue measure we obtain convergence also in the Sobolev norm. Keywords: Metric space, doubling measure, Poincaré inequality, Newtonian space, Harnack inequality, Harnack convergence theorem Mathematics Subject Classification (2000): 49J52, 35J60, 49J27     

Newton’s method on generalized Banach spaces

We present a weaker convergence analysis of Newton’s method than in Kantorovich and Akilov (1964), Meyer (1987), Potra and Ptak (1984), Rheinboldt (1978), Traub (1964) on a generalized Banach space setting to approximate a locally unique zero of an operator. This way we extend the applicability of Newton’s method. Moreover, we obtain under the same conditions in the semilocal case weaker sufficient convergence criteria; tighter error bounds on the distances involved and an at least as precise information on the location of the solution. In the local case we obtain a larger radius of convergence and higher error estimates on the distances involved. Numerical examples illustrate the theoretical results.     

How to improve the domain of parameters for Newton’s method

We study the influence of a center Lipschitz condition for the first derivative of the operator involved when the solution of a nonlinear equation is approximated by Newton’s method in Banach spaces. As a consequence, we see that the domain of parameters associated to the Newton–Kantorovich theorem is enlarged.     

A variant of Newton’s method based on Simpson’s three-eighths rule for nonlinear equations

We propose a new variant of Newton’s method based on Simpson’s three-eighth rule. It can be shown that the new method is cubically convergent.     

The radius of attraction for Newton’s method and TV-minimization image denoising

This paper presents an estimate of the radius of convergence for Newton’s method applied to a regularized TV-minimization problem often employed to denoise images. Strong Fréchet differentiability of this iteration is established and an estimate of the radius of attraction for this method is shown.     

Newton’s Method for M-Tensor Equations

Journal of Optimization Theory and Applications - We are concerned with the tensor equations whose coefficient tensors are M-tensors. We first propose a Newton method for solving the equation with...     

Newton’s method for generalized equations: a sequential implicit function theorem

In an extension of Newton’s method to generalized equations, we carry further the implicit function theorem paradigm and place it in the framework of a mapping acting from the parameter and the starting point to the set of all associated sequences of Newton’s iterates as elements of a sequence space. An inverse function version of this result shows that the strong regularity of the mapping associated with the Newton sequences is equivalent to the strong regularity of the generalized equation mapping.     

Improved estimates on majorizing sequences for the Newton–Kantorovich method

We approximate a locally unique solution of an equation in Banach space using the Newton–Kantorovich method. Motivated by our earlier works (see references [2–7] in the references list), optimization consideration, and the elegant studies by Cianciaruso with DePascale in (Numer. Funct. Anal. Optim. 27(5–6):529–538, 2006), and Cianciaruso in (Nonlinear Funct. Anal. Appl., 2009, to appear), we provide (by using more precise error estimates on the distances involved): finer error bounds; an at least as precise information on the location of the solution, and a larger convergence domain than in (Numer. Funct. Anal. Optim. 27(5–6):529–538, 2006). Finally, we provide numerical examples where our results can apply to solve equations, but earlier ones can not (see references [8–19]).     

Stolarsky’s invariance principle for projective spaces

We show that Stolarsky’s invariance principle, known for point distributions on the Euclidean spheres, can be extended to the real, complex, and quaternionic projective spaces and the octonionic projective plane.     

Modifications of Newton’s method for even-grade palindromic polynomials and other twined polynomials

The paper describes some modifications of Newton??s method for refining the zeros of even-grade f(x)-twined (f(x)-egt) polynomials, defined as polynomials whose roots appear in pairs {x _i ,f(x _i )}. Particular attention is given to even-grade palindromic (egp) polynomials. The algorithms are derived from certain symmetric division processes for computing a symmetric quotient and a symmetric remainder of two given f(x)-egt polynomials. Numerical results indicate that the presented algorithms can be more accurate than other methods which do not take into consideration the symmetry of the coefficients.     

Improved two-step Newton’s method for computing simple multiple zeros of polynomial systems

Numerical Algorithms - Given a polynomial system f that is associated with an isolated singular zero ξ whose Jacobian matrix is of corank one, and an approximate zero x that is close to...     

General convergence conditions of Newton’s method for m-Fréchet differentiable operators

We present a local as well as a semilocal convergence analysis for Newton’s method for approximating a locally unique solution of a nonlinear equation in a Banach space setting. Our hypotheses involve m-Fréchet-differentiable operators and general Lipschitz-type hypotheses, where m≥2 is a positive integer. The new convergence analysis unifies earlier results; it is more flexible and provides a finer convergence analysis than in earlier studies such as Argyros in J. Comput. Appl. Math. 131:149–159, 2001, Argyros and Hilout in J. Appl. Math. Comput. 29:391–400, 2009, Argyros and Hilout in J. Complex. 28:364–387, 2012, Argyros et al. Numerical Methods for Equations and Its Applications, CRC Press/Taylor & Francis, New York, 2012, Gutiérrez in J. Comput. Appl. Math. 79:131–145, 1997, Ren and Argyros in Appl. Math. Comput. 217:612–621, 2010, Traub and Wozniakowski in J. Assoc. Comput. Mech. 26:250–258, 1979. Numerical examples are presented further validating the theoretical results.     

Global convergence of Newton’s method on an interval

The solution of an equation f(x)= given by an increasing function f on an interval I and right-hand side , can be approximated by a sequence calculated according to Newtons method. In this article, global convergence of the method is considered in the strong sense of convergence for any initial value in I and any feasible right-hand side. The class of functions for which the method converges globally is characterized. This class contains all increasing convex and increasing concave functions as well as sums of such functions on the given interval. The characterization is applied to Keplers equation and to calculation of the internal rate of return of an investment project.An earlier version was presented at the Joint National Meeting of TIMS and ORSA, Las Vegas, May 7–9, 1990. Financial support from Økonomisk Forskningsfond, Bodø, Norway, is gratefully acknowledged. The author thanks an anonymous referee for helpful comments and suggestions.