On interval predictor-corrector methods

One can approximate numerically the solution of the initial value problem using single or multistep methods. Linear multistep methods are used very often, especially combinations of explicit and implicit methods. In floating-point arithmetic from an explicit method (a predictor), we can get the first approximation to the solution obtained from an implicit method (a corrector). We can do the same with interval multistep methods. Realizing such interval methods in floating-point interval arithmetic, we compute solutions in the form of intervals which contain all possible errors. In this paper, we propose interval predictor-corrector methods based on conventional Adams-Bashforth-Moulton and Nyström-Milne-Simpson methods. In numerical examples, these methods are compared with interval methods of Runge-Kutta type and methods based on high-order Taylor series. It appears that the presented methods yield comparable approximations to the solutions.     

Decomposition and Dynamic Cut Generation in Integer Linear Programming

Decomposition algorithms such as Lagrangian relaxation and Dantzig-Wolfe decomposition are well-known methods that can be used to generate bounds for mixed-integer linear programming problems. Traditionally, these methods have been viewed as distinct from polyhedral methods, in which bounds are obtained by dynamically generating valid inequalities to strengthen an initial linear programming relaxation. Recently, a number of authors have proposed methods for integrating dynamic cut generation with various decomposition methods to yield further improvement in computed bounds. In this paper, we describe a framework within which most of these methods can be viewed from a common theoretical perspective. We then discuss how the framework can be extended to obtain a decomposition-based separation technique we call decompose and cut. As a by-product, we describe how these methods can take advantage of the fact that solutions with known structure, such as those to a given relaxation, can frequently be separated much more easily than arbitrary real vectors.     

大系统试验选优理论研究

针对某些高维动态、大型线性、复杂非线性、复杂模拟仿真和定性知识模型，该文全面总结了作者采用数学模型、知识模型和试验选优理论相结合的方法而提出的试验选优理论，该理论体系可以使一些目前无法求解或很难求解的大型问题获得近似解。并在复杂水资源系统的优化中取得了良好的效果。     

Nonlinear Schwarz iterations with reduced rank extrapolation

Extrapolation methods can be a very effective technique used for accelerating the convergence of vector sequences. In this paper, these methods are used to accelerate the convergence of Schwarz iterative methods for nonlinear problems. A new implementation of the reduced-rank-extrapolation (RRE) method is introduced. Some convergence analysis is presented, and it is shown numerically that certain extrapolation methods can indeed be very effective in accelerating the convergence of Schwarz methods.     

Second order Chebyshev methods based on orthogonal polynomials

Summary. Stabilized methods (also called Chebyshev methods) are explicit Runge-Kutta methods with extended stability domains along the negative real axis. These methods are intended for large mildly stiff problems, originating mainly from parabolic PDEs. The aim of this paper is to show that with the use of orthogonal polynomials, we can construct nearly optimal stability polynomials of second order with a three-term recurrence relation. These polynomials can be used to construct a new numerical method, which is implemented in a code called ROCK2. This new numerical method can be seen as a combination of van der Houwen-Sommeijer-type methods and Lebedev-type methods. Received January 14, 2000 / Revised version received November 3, 2000 / Published online May 4, 2001     

Multi-modal method for chatter stability prediction and control in milling of thin-walled workpiece

Chatter stability in milling can be predicted by analytical methods or numerical methods. The system should be considered as multi-modal in milling of thin-walled workpiece. This paper proposes a numerical difference method based on Adams-Bashforth scheme. Moreover, multi-modal scheme of numerical methods is proposed. Analytical methods and numerical methods are verified by performing a series of milling trials. The experimental results are consistent with the predicted critical stability boundaries. Moreover, a new method for analyzing the computational time of analytical methods and numerical methods, which is based on the time complexity modeling, is presented. Computational time can be expressed as exact mathematical expression. By using the expression, the rate of increase of computational time can be derived.     

综合评价大学生的方法研究

本文通过实例研究了评价大学生的平均成绩排序法、蜘网图评价法和系统聚类分析法，通过这些方法，既可以得到学生个体在群体当中的名次，又可知道其个体特征，还可以对学生群体进行分类，从而获得综合评价大学生的结果     

A simply constructed third-order modifications of Newton's method

In this paper, we present a simple and easily applicable approach to construct some third-order modifications of Newton's method for solving nonlinear equations. It is shown by way of illustration that existing third-order methods can be employed to construct new third-order iterative methods. The proposed approach is applied to the classical Chebyshev–Halley methods to derive their second-derivative-free variants. Numerical examples are given to support that the methods thus obtained can compete with known third-order methods.     

LINEAR SYSTEMS ASSOCIATED WITH NUMERICAL METHODS FOR CONSTRAINED OPITMIZATION

Linear systems associated with numerical methods for constrained optimization are discussed in this paper. It is shown that the corresponding subproblems arise in most well-known methods, no matter line search methods or trust region methods for constrained optimization can be expressed as similar systems of linear equations. All these linear systems can be viewed as some kinds of approximation to the linear system derived by the Lagrange-Newton method. Some properties of these linear systems are analyzed.     

Variational iteration technique for solving nonlinear equations

In this paper, we use the variational iteration technique to suggest some new iterative methods for solving nonlinear equations f(x)=0. We also discuss the convergence criteria of these new iterative methods. Comparison with other similar methods is also given. These new methods can be considered as an alternative to the Newton method. We also give several examples to illustrate the efficiency of these methods. This technique can be used to suggest a wide class of new iterative methods for solving system of nonlinear equations.