Newton's method on the complex exponential function

We show that when Newton's method is applied to the product of a polynomial and the exponential function in the complex plane, the basins of attraction of roots have finite area.

     

一类整函数的牛顿法的拟直接盘和直接盘

In this paper, we consider Newton＇s method for a class of entire functions with infinite order. By using theory of dynamics of functions meromorphic outside a small set, we find there are some series of virtual immediate basins in which the dynamics converges to infinity and a series of immediate basins with finite area in the Fatou sets of Newton＇s method.     

Diophantine方程组a~2+b~2=c~r和x~2+b~y=c~z的一点注记

设r是大于1的正奇数,a,b,c是满足a~2+b~2=c~r的互素正整数.证明了:当r(?)5(mod8),c>10~(12)r~4且b是奇素数的方幂时,方程x~2+b~y=c~z仅有正整数解(x,y,z)=(a,2,r).     

Rate of convergence of a generalization of Newton's method

The Newton's method for finding the root of the equation (t)=0 can be easily generalized to the case where is monotone, convex, but not differentiable. Then, the convergence is superlinear. The purpose of this note is to show that the convergence is only superlinear. Indeed, for all (1, 2), we exhibit an example where the convergence of the iterates is exactly .     

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.     

牛顿变换Julia集的对称性

主要讨论多项式的牛顿变换Julia集的对称性问题．利用复动力系统理论,证明了多项式P（z）的Julia集的对称群是其牛顿变换Np（z）的Julia集的对称群的子群．获得了Julia集为一水平直线的充分必要条件．     

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.     

ODE-IVP-PACK via Sinc indefinite integration and Newton's method

This paper describes a package of computer programs for the unified treatment of initial-value problems for systems of ordinary differential equations. The programs implement a numerical method which is efficient for a general class of differential equations. The user may determine the solutions over finite or infinite intervals. The solutions may have singularities at the end-points of the interval for which the solution is sought. Besides giving the initial values and the analytical expression for the differential equations to be solved the user needs to specify the nature of the singularities and give some other analytical information as described in the paper in order to take advantage of the speed and accuracy of the package described here. This revised version was published online in June 2006 with corrections to the Cover Date.     

One-step jackknife for M-estimators computed using Newton's method

To estimate the dispersion of an M-estimator computed using Newton's iterative method, the jackknife method usually requires to repeat the iterative process n times, where n is the sample size. To simplify the computation, one-step jackknife estimators, which require no iteration, are proposed in this paper. Asymptotic properties of the one-step jackknife estimators are obtained under some regularity conditions in the i.i.d. case and in a linear or nonlinear model. All the one-step jackknife estimators are shown to be asymptotically equivalent and they are also asymptotically equivalent to the original jackknife estimator. Hence one may use a dispersion estimator whose computation is the simplest. Finite sample properties of several one-step jackknife estimators are examined in a simulation study.The research was supported by Natural Sciences and Engineering Research Council of Canada.     

商高数的Je?manowícz猜想

利用Bilu、Hanrot 和 Voutier关于本原素因子存在性理论及二次丢番图方程解的表示方面的一些精细结果证明:当a=n+1, b=2n(n+1), c=2n(n+1)+1时, 方程a^x+b^y=c^z仅有正整数解(x,y,z)=(2,2,2).       

A remark on Jeśmanowicz’ conjecture for the non-coprimality case

Let a, b, c be relatively prime positive integers such that a² + b² = c². Je?manowicz' conjecture on Pythagorean numbers states that for any positive integer N, the Diophantine equation (aN)^x + (bN)^y = (cN)^z has no positive solution (x, y, z) other than x = y = z = 2. In this paper, we prove this conjecture for the case that a or b is a power of 2.     

关于Lucas三角形猜想的一个结果

运用初等方法,证明k=7时Lucas三角形不存在.     

Extending the choice of starting points for Newton's method

In this paper, we propose a center Lipschitz condition for the second derivative together with the use of restricted domains in order to improve the starting points for Newton's method when compared with previous results. Moreover, we present some numerical examples validating the theoretical results.     

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.     

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.     

A variant of Newton's method with accelerated third-order convergence

In the given method, we suggest an improvement to the iteration of Newton's method. Derivation of Newton's method involves an indefinite integral of the derivative of the function, and the relevant area is approximated by a rectangle. In the proposed scheme, we approximate this indefinite integral by a trapezoid instead of a rectangle, thereby reducing the error in the approximation. It is shown that the order of convergence of the new method is three, and computed results support this theory. Even though we have shown that the order of convergence is three, in several cases, computational order of convergence is even higher. For most of the functions we tested, the order of convergence in Newton's method was less than two and for our method, it was always close to three.     

Peters type polynomials and numbers and their generating functions: Approach with p‐adic integral method

The aim of this paper is to introduce and investigate some of the primary generalizations and unifications of the Peters polynomials and numbers by means of convenient generating functions and p‐adic integrals method. Various fundamental properties of these polynomials and numbers involving some explicit series and integral representations in terms of the generalized Stirling numbers, generalized harmonic sums, and some well‐known special numbers and polynomials are presented. By using p‐adic integrals, we construct generating functions for Peters type polynomials and numbers (Apostol‐type Peters numbers and polynomials). By using these functions with their partial derivative eqautions and functional equations, we derive many properties, relations, explicit formulas, and identities including the Apostol‐Bernoulli polynomials, the Apostol‐Euler polynomials, the Boole polynomials, the Bernoulli polynomials, and numbers of the second kind, generalized harmonic sums. A brief revealing and historical information for the Peters type polynomials are given. Some of the formulas given in this article are given critiques and comments between previously well‐known formulas. Finally, two open problems for interpolation functions for Apostol‐type Peters numbers and polynomials are revealed.     

Generalized differentiability conditions for Newton's method

The use of majorizing sequences is the usual way to prove theconvergence of Newton's method. An alternative technique tomajorizing sequences is provided in this paper, in which threescalar sequences are used, so that the analysis of convergenceis simplified when the traditional convergence condition isrelaxed. An application to a nonlinear integral equation isalso given, which is also solved and the solution approximatedby a discretization process.