有向图在求解大型非线性方程组中的应用

为了提高大维数非线性方程组的求解品质,该文提出了有向图法,将参数关系转化为有向图,利用有向图求圈算法来精简迭代参数,从而提高计算速度,降低算法发散的可能性。这种方法适用于求解工程化的大维数非线性方程组,可以用计算机自动实现。     

一种自学习模式参数化设计方法的研究

本文提出一种和一般参数化方法有本质区别的自学习模式的智能参数化设计方法,首先讨论了基于神经网络的参数化自学习原理与约束方程组求解的相似性和可行性,在分析了一种径向基函数网络的算法原理和网络结构后,重点探讨了应用自学习方式的参数化建模过程,并给出它在工程图参数化设计中的应用方法和特点.     

球面坐标下的凸组合球面参数化

球面参数化是一种应用价值很广的几何参数化方法.对于封闭且亏格为零的三角形网格,该文提出了一种新的球面参数化方法.通过引入多个球面坐标覆盖,在球面坐标系下,用凸组合方法,得到了接近线性的球面参数化求解方法.与已有的直角坐标系下的凸组合参数化方法相比,该文所提出的方法大大降低了求解方程组的非线性程度,因此求解时间大幅度降低.此外,还避免了直角坐标系下求解的多种退化情况.最后,给出了实验结果,并对凸组合球面参数化中存在的几个问题进行了讨论.     

基于规则求解的约束驱动的变量几何造型系统

本文系统地阐述了一种在进行几何型体设计时用约束关系和工程关系来驱动几何型体的方法。提出了对约束方程组进行分解的算法和在规则求解的基础上分两层求解方程组的方法。关键词     

一种求解摄像机内参数的非线性方法

提出了一种求解摄像机内参数的非线性方法。首先对空间单个矩形的图像获得含参数，的圆环点的图像，其中r为该矩形的长宽比；进而由圆环点对摄像机内参数的约束获得一组非线性方程；通过把非线性方程组求解问题转为含3个参数的最优化问题来确定摄像机的内参数。该方法的特点在于，无需知道矩形的任何信息，也容易将所获得的非线性方程组转化为最优化问题。实验结果表明，该方法具有较高的求解精度。     

电磁逆散射成像的一种混合正则化方法

电磁逆散射成像问题数值求解中,非线性逆散射方程及其对应的离散方程组具有明显的不适定性,针对求解通常所用Tikhonov正则化方法的参数选择在先验选取时缺乏有效的误差信息,而后验选取时需要更多计算量求解有关参数的方程的困难,本文中将小参数Tikhonov正则化方法与共轭梯度法结合,提出了不适定方程组的混合正则化方法。数据仿真表明,该方法既可保证正则化效果,也减少了计算量。     

层次泛函网络整体学习算法

文中设计了一类单输人单输出泛函网络与双输人单输出泛函网络作为构造层次泛函网络基本模型,提出了一种层次泛函网络模型,给出了层次泛函网络构造方法和整体学习算法,而层次泛函网络的参数利用解方程组来进行逐层学习．以非线性代数方程组为例,指出人们熟知的一些数学解题方法可以用层次泛函网络来表达,探讨了基于层次泛函网络求解非线性代数方程组学习算法实现的一些技术问题．相对传统方法,层次泛函网络更适合于具有层次结构的应用领域．计算机仿真结果表明,这种层次学习方法具有较快的收敛速度和良好的逼近性能．     

正定矩阵支持向量机正则化路径算法

正则化路径算法是数值求解支持向量机 (support vector machine, SVM)分类问题的有效方法,它可在相当于一次SVM求解的时间复杂度内得到所有的正则化参数及对应SVM的解.现有的SVM正则化路径算法或者不能处理具有重复数据、近似数据或线性相关数据,或者计算开销较大.针对这些问题,应用正定矩阵方程组求解方法来求解SVM正则化路径,提出正定矩阵SVM正则化路径算法(positive definite SVM path, PDSVMP).PDSVMP算法将迭代方程组的系数矩阵转换为正定矩阵,并采用Cholesky分解方法求解路径上各拐点处Lagrange乘子增量向量;与已有算法中直接求解正则化参数不同,该算法根据活动集变化情况确定参数增量,并在此基础上计算正则化参数,这样保证了理论正确性和数值稳定性,并可降低计算复杂性.实例数据集及标准数据集上的实验表明,PDSVMP算法可正确处理包含重复数据、近似数据或线性相关数据的数据集,并具有较高的计算效率.     

基于约束的几何设计系统原理及实现

本文提出了一种基于约束的几何设计系统的基本原理和方法。通过对几何约束系统的最大分解，使整个系统的求解可以通过一系列子系统的求解实现。实践表明，该方法大大缩小了问题求解的规模，提高了求解精度和速度，为参数化设计提供了强有力的手段。     

多维Newton—Moser算法的一种改进

一、引言在控制工程、计算机仿真及最优化设计等领域,往往要求解非线性方程。但多数方程不能用解析方法求解,即使可以求解也是相当困难的。因而我们常常把方程变换为离散化的线性方程组或非线性方程组,然后选用适当的数值算法在计算机上求解。求解线性方程组的算法很多,此文不予讨论;求解非线性方程组的算法虽多,但许多算法在应     

Application of Sinc-collocation method for solving a class of nonlinear Fredholm integral equations

In this paper, the numerical solution of nonlinear Fredholm integral equations of the second kind is considered by two methods. The methods are developed by means of the Sinc approximation with the single exponential (SE) and double exponential (DE) transformations. These numerical methods combine a Sinc collocation method with the Newton iterative process that involves solving a nonlinear system of equations. We provide an error analysis for the methods. So far approximate solutions with polynomial convergence have been reported for this equation. These methods improve conventional results and achieve exponential convergence. Some numerical examples are given to confirm the accuracy and ease of implementation of the methods.     

计算微分代数系统的实时仿真算法

§１．引言 对于由微分代数方程所表示的动力系统的数值算法,针对微分代数方程的一些特殊形式已经构造了一些有效算法如文献[1]-［４］．这些数值算法大部分都是基于常微分方程的一些隐式公式如隐式Ｒｕｎｇｅ－Ｋｕｔｔａ方法,向后微分公式（ＢＤＦ）等,因此这些算法都是非实时仿真算法．如果我们直接用求解常微分方程的显式公式如显式 Ｒｕｎｇｅ－Ｋｕｔｔａ方法,显式线性多步法等,虽然满足了实时仿真算法的一些特点,但是这些数值公式对微分代数方程的求解不甚理想．由于一个实时仿真算法具有实时性、周期性、可靠性等特性要求,因…     

含时滞的半主动系统动力学分析

对含时滞的半主动相对控制悬架系统进行了近似解析研究.首先建立了半主动相对控制1/4车体模型,进行了无量纲化处理,利用平均法建立了系统的近似解析解应该满足的四元代数方程组,然后利用数值方法进行了求解.随后通过MATLAB仿真得到了含时滞的半主动相对控制悬架系统的数值解,并且和近似解析解进行了比较,发现二者具有较好的符合精度,说明近似解析解的正确性.     

Convergence of the Euler–Maruyama method for stochastic differential equations with Markovian switching

Stochastic differential equations with Markovian switching (SDEwMSs), one of the important classes of hybrid systems, have been used to model many physical systems that are subject to frequent unpredictable structural changes. The research in this area has been both theoretical and applied. Most of SDEwMSs do not have explicit solutions so it is important to have numerical solutions. It is surprising that there are not any numerical methods established for SDEwMSs yet, although the numerical methods for stochastic differential equations (SDEs) have been well studied. The main aim of this paper is to develop a numerical scheme for SDEwMSs and estimate the error between the numerical and exact solutions. This is the first paper in this direction and the emphasis lies on the error analysis.     

Convergence and stability of numerical methods with variable step size for stochastic pantograph differential equations

The stochastic pantograph equations (SPEs) are very special stochastic delay differential equations (SDDEs) with unbounded memory. When the numerical methods with a constant step size are applied to the pantograph equations, the most difficult problem is the limited computer memory. In this paper, we construct methods with variable step size to solve SPEs. The analysis is motivated by the example of a mean-square stable linear SPE for which the Euler–Maruyama (EM) method with variable step size fails to reproduce this behaviour for any nonzero timestep. Then we consider the Backward Euler (BE) method with variable step size and develop the fundamental numerical analysis concerning its strong convergence and mean-square linear stability. It is proved that the numerical solutions produced by the BE method with variable step size converge to the exact solution under the local Lipschitz condition and the Bounded condition. Furthermore, the order of convergence p=½ is given under the Lipschitz condition. The result of the mean-square linear stability is given. Some illustrative numerical examples are presented to demonstrate the order of strong convergence and the mean-square linear stability of the BE method.     

An iterative splitting method via waveform relaxation

This paper explores a new numerical strategy for a closed formulation of iterative splitting methods and their embedding in classical waveform-relaxation methods. Since iterative splitting has been developed in several papers, an abstract framework that relates these methods to other classical splitting methods would be useful and is needed. Here, we present an embedding of the iterative splitting method in the waveform-relaxation and exponential splitting methods. While we can use the theoretical background of the classical schemes, a simpler iterative splitting analysis is obtained. This is achieved by basing the analysis on semigroup and fixpoint schemes. Our approach is illustrated with numerical results obtained on differential equations with constant and time-dependent coefficients.     

The qualitative and semiqualitative analysis of environmental problems

The analysis of many biochemical engineering problems in environmental modelling is based upon the development and solution of sets of differential equations. A complete analytical solution of such a model requires that every numerical constant in this set of equations is precisely known. This paper describes the use of methods from artificial intelligence which permit the solution of such sets of equations when some constant values are unknown. The use of the methods are illustrated with the solution of a set of equations representing one model of an anaerobic fermentor, and a computer program that implements the methods is described.     

Collocation method for solving systems of Fredholm and Volterra integral equations

In this paper, a numerical method based on based quintic B-spline has been developed to solve systems of the linear and nonlinear Fredholm and Volterra integral equations. The solutions are collocated by quintic B-splines and then the integral equations are approximated by the four-points Gauss-Turán quadrature formula with respect to the weight function Legendre. The quintic spline leads to optimal approximation and O(h⁶) global error estimates obtained for numerical solution. The error analysis of proposed numerical method is studied theoretically. The results are compared with the results obtained by other methods which show that our method is accurate.     

Seismic analysis of base isolated buildings

Summary  This paper presents a survey of the numerical simulation of base isolation systems for the vibration control of buildings and their equipment, primarilly against earthquakes. Base isolation has received much attention in the recent twenty years and many buildings have been protected using this technology. The article focusses mainly on the different numerical methods used in the analysis of base isolated buildings. The conventional form of solving the equations of motion governing the seismic response of building structures with nonlinear base isolation consists of using monolithic step by step integration methods. As an efficient alternative static condensation and block iterative schemes can be applied. The particularities of the equations of motion of buildings equiped with various base isolation systems are described. The linear theory of base isolated buildings is then presented. After this, numerical solution techniques for the analysis of the seismic response of buildings with isolation systems are developed in detail in the paper. Finally, numerical results for elastic and inelastic structures are described. A complete set of references coverning a wide range of studies is included.     

Dynamics analysis of linear elastic planar mechanisms

This paper presents a formulation for the dynamics analysis of an elastic mechanism and integrating a stiff system using efficient numerical methods. Because all the elastic degrees of freedom are included in the vector of generalized variables, the size of the equations is much larger than that obtained using either the assumed mode or the distributed parameter finite element approach. However, the resulting system matrix is sparse and the elastic coordinates are absent from the system matrix, and these are useful properties for subsequent numerical analysis. Techniques for solving a system of linear time-variant equations are applied to the dynamics equations, assuming that the system matrix is slow-changing, and thus, may be approximated by a series of piecewise constant matrices. It is argued that the problem of determining the integration time step is transformed into the problem of computing the exponential of the system matrix with automatic time scaling. A numerical example is given to show that the behavior of the rigid coordinates converges to that of an all-rigid-body model by artificially increasing the Young’s modulus of the elastic components, despite the very-high-frequency vibrations of the elastic coordinates induced by the increment of the stiffness.