On Newton's method for entire functions

The Newton map N_f of an entire function f turns the roots off into attracting fixed points. Let U be the immediate attractingbasin for such a fixed point of N_f. We study the behavior ofN_f in a component V of \U. If V can be surrounded by an invariantcurve within U and satisfies the condition that for all z , N_f^–1({z}) V is a finite set,then it is shown that V contains another immediate basin ofN_f or a virtual immediate basin.     

On Newton's method for solving generalized equations

In this paper, we study the convergence properties of a Newton-type method for solving generalized equations under a majorant condition. To this end, we use a contraction mapping principle. More precisely, we present semi-local convergence analysis of the method for generalized equations involving a set-valued map, the inverse of which satisfying the Aubin property. Our analysis enables us to obtain convergence results under Lipschitz, Smale and Nesterov-Nemirovski's self-concordant conditions.     

Leap-frogging Newton's method

Using Newton's method as an intermediate step, we introduce an iterative method that approximates numerically the solution of f (x) = 0. The method is essentially a leap-frog Newton's method. The order of convergence of the proposed method at a simple root is cubic and the computational efficiency in general is less, but close to that of Newton's method. Like Newton's method, the new method requires only function and first derivative evaluations. The method can easily be implemented on computer algebra systems where high machine precision is available.     

On the dynamics of a third order Newton's approximation method

We provide an answer to a question raised by S. Amat, S. Busquier, S. Plaza on the qualitative analysis of the dynamics of a certain third order Newton type approximation function , by proving that for functions f twice continuously differentiable and such that both f and its derivative do not have multiple roots, with at least four roots and infinite limits of opposite signs at , has periodic points of any prime period and that the set of points a at which the approximation sequence does not converge is uncountable. In addition, we observe that in their Scaling Theorem analyticity can be replaced with differentiability.     

On Newton's interpolation polynomials

In this paper functional analytic methods for nuclear locally convex spaces are applied to problems of analytic functions. The question is discussed whether the so-called Newton interpolation polynomials constitute a Schauder-basis in the space of analytic functions on the open unit circle (see Marku evi [3]). There are several different approaches to this problem, see, for instance, Walsh [7] and Gelfond [1]. Here we give a necessary and sufficient condition in terms of the interpolation points only. We consider the above space of analytic functions as a nuclear Kö-the-sequence space and use some deep theorems about nuclear spaces, such as the theorem of Dynin and Mitjagin (see Rolewicz [6], Pietsch An interesting connection with the theory of uniformly distributed sequences is mentioned.     

Newton's method for nonlinear inequalities

A Newton-type algorithm has been presented elsewhere for solving non-linear inequalities of the formf(x)0,g(x)=0, and quadratic convergence has been proved under very strong hypotheses. In this paper we show that the same results hold under a considerable weakening of the hypotheses.This research was supported in part by contract number N00014-67-A-0126-0015, NR 044-425, from the Office of Naval Research.     

Third-order modification of Newton's method

In this paper, we present a new modification of Newton's method for solving non-linear equations. Analysis of convergence shows that the new method is cubically convergent. Numerical examples show that the new method can compete with the classical Newton's method.     

Newton's method for fractal approximation

The problem of fitting a given function in theL ^q norm with a function generated by an iterated function system can be rapidly solved by applying Newton's method on the parameter space of the iterated function system. The key to this is a method for calculating the derivatives of a potential function with respect to the parameters.     

Newton's method for constrained optimization

We derive a quadratically convergent algorithm for minimizing a nonlinear function subject to nonlinear equality constraints. We show, following Kaufman [4], how to compute efficiently the derivative of a basis of the subspace tangent to the feasible surface. The derivation minimizes the use of Lagrange multipliers, producing multiplier estimates as a by-product of other calculations. An extension of Kantorovich's theorem shows that the algorithm maintains quadratic convergence even if the basis of the tangent space changes abruptly from iteration to iteration. The algorithm and its quadratic convergence are known but the drivation is new, simple, and suggests several new modifications of the algorithm.     

On the local classification of smooth maps induced by Newton's method

The classification of germs of smooth maps f:Rⁿ→Rⁿ induced by the continuous Newton method is addressed in this paper. This classification problem is shown to rely on an equivalence notion located between right and contact equivalences, driving the classification problem from the setting of quasilinear ODEs to the singularity theory framework. One-dimensional problems and regular, n-dimensional cases are easily characterized, and normal forms for them are given. Folded zeros in Rⁿ display a much richer behavior: an invariant and a preliminary normal form are derived for these cases.     

On the convergence of Newton's method when estimating higher dimensional parameters

In this paper, we consider the estimation of a parameter of interest where the estimator is one of the possibly several solutions of a set of nonlinear empirical equations. Since Newton's method is often used in such a setting to obtain a solution, it is important to know whether the so obtained iteration converges to the locally unique consistent root to the aforementioned parameter of interest. Under some conditions, we show that this is eventually the case when starting the iteration from within a ball about the true parameter whose size does not depend on n. Any preliminary almost surely consistent estimate will eventually lie in such a ball and therefore provides a suitable starting point for large enough n. As examples, we will apply our results in the context of M-estimates, kernel density estimates, as well as minimum distance estimates.     

On Newton's method for Huber's robust M-estimation problems in linear regression

The Newton method of Madsen and Nielsen (1990) for computing Huber's robust M-estimate in linear regression is considered. The original method was proved to converge finitely for full rank problems under some additional restrictions on the choice of the search direction and the step length in some degenerate cases. It was later observed that these requirements can be relaxed in a practical implementation while preserving the effectiveness and even improving the efficiency of the method. In the present paper these enhancements to the original algorthm are studied and the finite termination property of the algorithm is proved without any assumptions on the M-estimation problems. Research supported by NATO Collaborative Research Grant CRG-94-0609.     

An acceleration of Newton's method: Super-Halley method

From a study of the convexity we give an acceleration for Newton's method and obtain a new third order method. Then we use this method for solving non-linear equations in Banach spaces, establishing conditions on convergence, existence and uniqueness of solution, as well as error estimates     

A nonsmooth version of Newton's method

Newton's method for solving a nonlinear equation of several variables is extended to a nonsmooth case by using the generalized Jacobian instead of the derivative. This extension includes the B-derivative version of Newton's method as a special case. Convergence theorems are proved under the condition of semismoothness. It is shown that the gradient function of the augmented Lagrangian forC ²-nonlinear programming is semismooth. Thus, the extended Newton's method can be used in the augmented Lagrangian method for solving nonlinear programs.This author's work is supported in part by the Australian Research Council.This author's work is supported in part by the National Science Foundation under grant DDM-8721709.     

Newton's method in generalized banach spaces

A Kantorowitsch-analysis of Newton's method in generalized Banach spaces is given. The application of generalized norms – mappings from a linear space into a partially ordered Banach space - improves convergence results and error estimatescompared with the real norm theory.     

Newton's method for linear complementarity problems

This paper presents an iterative, Newton-type method for solving a class of linear complementarity problems. This class was discovered by Mangasarian who had established that these problems can be solved as linear programs. Cottle and Pang characterized solutions of the problems in terms of least elements of certain polyhedral sets. The algorithms developed in this paper are shown to converge to the least element solutions. Some applications and computational results are also discussed.     

Newton's method and complex dynamical systems

This article is devoted to the discussion of Newton's method. Beginning with the old results of A.Cayley and E.Schröder we proceed to the theory of complex dynamical systems on the sphere, which was developed by G.Julia and P.Fatou at the beginning of this century, and continued by several mathematicians in recent years.     

On the convergence of Newton's method for a class of nonsmooth operators

We provide an analog of the Newton–Kantorovich theorem for a certain class of nonsmooth operators. This class includes smooth operators as well as nonsmooth reformulations of variational inequalities. It turns out that under weaker hypotheses we can provide under the same computational cost over earlier works [S.M. Robinson, Newton's method for a class of nonsmooth functions, Set-Valued Anal. 2 (1994) 291–305] a semilocal convergence analysis with the following advantages: finer error bounds on the distances involved and a more precise information on the location of the solution. In the local case not examined in [S.M. Robinson, Newton's method for a class of nonsmooth functions, Set-Valued Anal. 2 (1994) 291–305] we can show how to enlarge the radius of convergence and also obtain finer error estimates. Numerical examples are also provided to show that in the semilocal case our results can apply where others [S.M. Robinson, Newton's method for a class of nonsmooth functions, Set-Valued Anal. 2 (1994) 291–305] fail, whereas in the local case we can obtain a larger radius of convergence than before [S.M. Robinson, Newton's method for a class of nonsmooth functions, Set-Valued Anal. 2 (1994) 291–305].