一个三阶牛顿变形方法

基于反函数建立的积分方程,结合Simpson公式,给出了一个非线性方程求根的新方法,即为牛顿变形方法.证明了它至少三次收敛到单根,与牛顿法相比,提高了收敛阶和效率指数.文末给出数值试验,且与牛顿法和同类型牛顿变形法做了比较.结果表明方法具有较好的优越性,它丰富了非线性方程求根的方法.     

Some modifications of Newton's method with fifth-order convergence

In this paper, we present some new modifications of Newton's method for solving non-linear equations. Analysis of convergence shows that these methods have order of convergence five. Numerical tests verifying the theory are given and based on these methods, a class of new multistep iterations is developed.     

一类新的求解非线性方程的七阶方法

利用权函数法给出了一类求解非线性方程单根的七阶收敛的方法.每步迭代需要计算三个函数值和一个导数值,因此方法的效率指数为1.627.数值试验给出了该方法与牛顿法及同类方法的比较,显示了该方法的优越性.最后指出Kou等人给出的七阶方法是方法的特例.     

Some variants of Cauchy's method with accelerated fourth-order convergence

In this paper, we present some variants of Cauchy's method for solving non-linear equations. Analysis of convergence shows that the methods have fourth-order convergence. Per iteration the new methods cost almost the same as Cauchy's method. Numerical results show that the methods can compete with Cauchy's method.     

一种新的改进Newton法(英文)

本文给出了求解非线性方程的一种新的改进方法.利用Newton法和Heron平均,将新改进方法与其它一些迭代法作比较.数值结果表明该方法具有一定的实用价值.     

A novel method for a class of parameterized singularly perturbed boundary value problems

In this paper a novel approach is presented for solving parameterized singularly perturbed two-point boundary value problems with a boundary layer. By the boundary layer correction technique, the original problem is converted into two non-singularly perturbed problems which can be solved using traditional numerical methods, such as Runge–Kutta methods. Several non-linear problems are solved to demonstrate the applicability of the method. Numerical experiments indicate the high accuracy and the efficiency of the new method.     

An Iterative method for solving fractional differential equations

In the present paper non-linear, time fractional advection partial differential equation has been solved using the new iterative method presented by Daftardar-Gejji and Jafari [1]. The results are compared with those obtained by Adomian decomposition and Homotopy perturbation methods. It is demonstrated that the new iterative method gives the best approximation among these. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

交通流灰色RBF网络非线性组合预测方法

针对智能交通系统的开发,提出一种基于灰色GM(1,1)模型和RBF网络非线性组合的短时交通流预测方法.该方法采用三层结构的RBF网络将两种单一预测方法(灰色GM(1,1)模型和RBF网络)进行了非线性组合.利用实测数据对组合方法进行了仿真实验,结果表明:非线性组合模型的预测准确性高于单独的RBF网络预测的准确性;组合模型发挥了两种单一方法各自的优势,是短时交通流预测的有效方法.     

Variational Methods for Non-Linear Least-Squares

We consider Newton-like line search descent methods for solving non-linear least-squares problems. The basis of our approach is to choose a method, or parameters within a method, by minimizing a variational measure which estimates the error in an inverse Hessian approximation. In one approach we consider sizing methods and choose sizing parameters in an optimal way. In another approach we consider various possibilities for hybrid Gauss-Newton/BFGS methods. We conclude that a simple Gauss-Newton/BFGS hybrid is both efficient and robust and we illustrate this by a range of comparative tests with other methods. These experiments include not only many well known test problems but also some new classes of large residual problem.     

求解非线性算子方程的加速迭代格式

研究非线性算子方程的近似求解方法.首先对通常的求解非线性方程加速迭代格式进行推广,得到高阶收敛速度的加速迭代格式,最后把这种加速迭代格式推广到非线性算子方程的求解中去,利用非线性算子的渐进展开,证明了这种加速格式具有三阶的收敛速度.     

On a Class of Non-Linear Methods for Ordinary Differential Equations

In this paper, a class of non-linear methods proposed in [1] is discussed. A new derivation of the methods is given. The analysis based on the new derivation shows that this class of methods is not suitable for stiff problems. The numerical tests support our argument.     

Solución de la cinemática directa de la plataforma Gough-Stewart general usando un algoritmo híbrido de optimización

The direct kinematics problem for parallel robots can be stated as follows: given values of the joint variables, the corresponding Cartesian variable values, the pose of the end-effector, must be found. Most of the times the direct kinematics problem involves the solution of a system of non-linear equations. The most efficient methods to solve such kind of equations assume convexity in a cost function which minimum is the solution of the non-linear system. In consequence, the capacity of such methods depends on the knowledge about an starting point which neighboring region is convex, hence the method can find the global minimum. This article propose a method based on probabilistic learning about an adequate starting point for the Dogleg method which assumes local convexity of the function. The proposed method efficiently avoids the local minima, without need of human intervention or apriori knowledge, thus it shows a more robust performance than the simple Dogleg method or other gradient based methods. To demonstrate the performance of the proposed hybrid method, numerical experiments and the respective discussion are presented. The proposal can be extended to other structures of closed-kinematics chains, to the general solution of systems of non-linear equations, and to the minimization of non-linear functions.     

解偏微分方程的多步-小波-Galerkin方法

推广Lax-Wendroff多步方法,建立一类新的显式和隐式相结合的多步格式,并以此为基础提出了一类显隐多步-小波-Galerkin方法,可以用来求解依赖时间的偏微分方程.不同于Taylor-Galerkin方法,文中的方案在提高时间离散精度时不包含任何新的高阶导数.由于引入了隐式部分,与传统的多步方法相比该方案有更好的稳定性,适合于求解非线性偏微分方程,理论分析和数值例子都说明了方法的有效性.     

A new family of Schröder's method and its variants based on power means for multiple roots of nonlinear equations

In this article, we derive one-parameter family of Schröder's method based on Gupta et al.'s (K.C. Gupta, V. Kanwar, and S. Kumar, A family of ellipse methods for solving non-linear equations, Int. J. Math. Educ. Sci. Technol. 40 (2009), pp. 571–575) family of ellipse methods for the solution of nonlinear equations. Further, we introduce new families of Schröder-type methods for multiple roots with cubic convergence. Proposed families are derived from modified Newton's method for multiple roots and one-parameter family of Schröder's method. Numerical examples are also provided to show that these new methods are competitive to other known methods for multiple roots.     

An asymptotic solution to non-linear transmission lines

This paper explores an asymptotic approach to the solution of a non-linear transmission line model. The model is based on a set of non-linear partial differential equations without analytical solution. The perturbations method is used to reduce the system of non-linear equations to a single non-linear partial differential equation, the modified Korteweg–de Vries equation (KdV). By using the Laplace transform, the solution is represented in integral form in terms of Green's functions. The solution for the non-linear case is obtained by means of asymptotic methods. Thus, an approximate explicit analytical solution to the problem is obtained where the errors can be controlled. This allows us to analyze the non-linear behavior of the solution. This kind of information is difficult to obtain by means of numerical methods due to the fact that for large periods of time greater computational resources are required and also accumulated errors increase. For this reason, asymptotic methods have a great importance like a natural complement to numerical methods. Computer simulations support the developments presented.     

Finding local optima of high-dimensional functions using direct search methods

This paper focuses on a subclass of box-constrained, non-linear optimization problems. We are particularly concerned with settings where gradient information is unreliable, or too costly to calculate, and the function evaluations themselves are very costly. This encourages the use of derivative free optimization methods, and especially a subclass of these referred to as direct search methods. The thrust of our investigation is twofold. First, we implement and evaluate a number of traditional direct search methods according to the premise that they should be suitable as local optimizers when used in a metaheuristic framework. Second, we introduce a new direct search method, based on Scatter Search, designed to remedy the lack of a good derivative free method for solving problems of high dimensions. Our new direct search method has convergence properties comparable to those of existing methods in addition to being able to solve larger problems more effectively.     

Variational iteration method for solving non-linear partial differential equations

In this paper, we shall use the variational iteration method to solve some problems of non-linear partial differential equations (PDEs) such as the combined KdV–MKdV equation and Camassa–Holm equation. The variational iteration method is superior than the other non-linear methods, such as the perturbation methods where this method does not depend on small parameters, such that it can fined wide application in non-linear problems without linearization or small perturbation. In this method, the problems are initially approximated with possible unknowns, then a correction functional is constructed by a general Lagrange multiplier, which can be identified optimally via the variational theory.     

Applications of information theory,genetic algorithms,and neural models to predict oil flow

This work introduces a new information-theoretic methodology for choosing variables and their time lags in a prediction setting, particularly when neural networks are used in non-linear modeling. The first contribution of this work is the Cross Entropy Function (XEF) proposed to select input variables and their lags in order to compose the input vector of black-box prediction models. The proposed XEF method is more appropriate than the usually applied Cross Correlation Function (XCF) when the relationship among the input and output signals comes from a non-linear dynamic system. The second contribution is a method that minimizes the Joint Conditional Entropy (JCE) between the input and output variables by means of a Genetic Algorithm (GA). The aim is to take into account the dependence among the input variables when selecting the most appropriate set of inputs for a prediction problem. In short, theses methods can be used to assist the selection of input training data that have the necessary information to predict the target data. The proposed methods are applied to a petroleum engineering problem; predicting oil production. Experimental results obtained with a real-world dataset are presented demonstrating the feasibility and effectiveness of the method.     

New third and fourth order nonlinear solvers for computing multiple roots

We present a new third order method for finding multiple roots of nonlinear equations based on the scheme for simple roots developed by Kou et al. [J. Kou, Y. Li, X. Wang, A family of fourth-order methods for solving non-linear equations, Appl. Math. Comput. 188 (2007) 1031-1036]. Further investigation gives rise to new third and fourth order families of methods which do not require second derivative. The fourth order family has optimal order, since it requires three evaluations per step, namely one evaluation of function and two evaluations of first derivative. The efficacy is tested on a number of relevant numerical problems. Computational results ascertain that the present methods are competitive with other similar robust methods.     

A modified combined relaxation method for non-linear convex variational inequalities

We consider a class of non-linear problems which is intermediate between equilibrium and variational inequality ones and has many applications. Unlike the usual variational inequality it involves two non-linear mappings, which need not be differentiable. We propose a class of iterative methods for this problem, which converge to a solution under weakened monotonicity type assumptions. This method is simpler essentially in comparison with those for the corresponding non-linear equilibrium problems.