由主子阵和缺损特征对构造Jacobi矩阵

１．引言设ｎ阶Ｊａｃｏｂｉ矩阵为 Ｊａｃｏｂｉ矩阵逆特征值问题的研究在振动工程、结构设计和系统参数识别等领域有重要应用．由主子阵和谱数据构造Ｊａｃｏｂｉ阵Ｊｎ,戴华首次得到ｎ为偶数时有解的充要条件,并给出了一个数值算法 ［１］;［２］对 ｎ为任意正整数时给出了一个新算法,此算法在计算过程中可自动判断解的存在性．由缺损特征对和谱数据构造三对角对称阵,［３］给出了有解的充要条件,本文研究由主子阵和缺损特征对构造Ｊａｃｏｂｉ矩阵,其问题如下： 问题Ａ．给定ｋ阶Ｊａｃｏｂｉ阵又给定和求和阶 Ｊａｃｏｂｉ阵使…     

延迟动力系统线性θ-方法的散逸性

１．引言 科学与工程中的许多问题具有散逸性,即系统具有一有界吸引集,从任意初始条件出发的解经过有限时间后进入该吸引集并随后保持在里面．如 ２维的 Ｎａｖｉｅｒ－Ｓｔｏｋｅｓ方程、Ｌｏｒｅｎｚ方程等许多重要系统都是散逸的．散逸性研究一直是动力系统研究中的重要课题（参见Ｔｅｍａｍ［７］）．当数值求解这些系统时,自然希望数值方法能保持系统的该重要特性．１９９４年, Ｈｕｍｐｈｒｉｅｓ和 Ｓｔｕａｒｔ［６］首次研究了 Ｒｕｎｇｅ－Ｋｕｔｔａ方法对有限维系统的散逸性．１９９７年Ｈｉｌｌ［２］研究了其无穷维散逸性…     

基于近似惯性流形的后验Galerkin的方法

１．引言 为提高用数值方法解非线性发展方程及非线性椭圆边值问题的逼近阶,许多学者例如Ｊ．Ｎｏｖｏ和 Ｅ．Ｔｉｔｉ［４］, Ｍａｒｉｏｎ和 Ｔｅｍａｎ［６］,Ｊ．Ｘｕ［７］以及 Ｗ．Ｌａｙｔｏｎ［９］等人,提出了后验Ｇａｌｅｒｋｉｎ方法、近似惯性流形方法、非线性Ｇａｌｅｒｋｉｎ方法、各种区域分裂法、多重网格法等等．本文根据［１］提出了一种新的高精度的后验 Ｇａｌｅｒｋｉｎ方法．它的逼近阶是经典 Ｇａｌｅｒｋｉｎ方法逼近阶的两倍． 考虑非线性椭圆边值问题这里ｎ是按ｄ＝２,３）上具有分段光滑边界ｒ的有界区域,…     

改进HS共轭梯度算法及其全局收敛性

１．引 言 １９５２年 Ｍ．Ｈｅｓｔｅｎｅｓ和Ｅ．Ｓｔｉｅｆｅｌ提出了求解正定线性方程组的共轭梯度法［１］．１９６４年Ｒ．Ｆｌｅｔｃｈｅｒ和Ｃ．Ｒｅｅｖｅｓ将该方法推广到求解下列无约束优化问题： ｍｉｎｆ（ｘ）,ｘ∈Ｒｎ,（１）其中ｆ：Ｒｎ→Ｒ１为连续可微函数,记ｇｋ＝ ｆ（ｘｋ）,ｘｋ∈ Ｒｎ． 若点列｛ｘｋ｝由如下算法产生：其中 βｋ＝［ｇＴｋ（ｇｋ－ｇｋ－１）］／［ｄＴｋ－１（ｇｋ－ｇｋ－１）］．（Ｈｅｓｔｅｎｅｓ－Ｓｔｉｅｆｅｌ）　　（４）则称该算法为 Ｈｅｓｔｅｎｅｓ—Ｓｔｉｅｆｅｌ共轭梯度算…     

一般线性方法的正则性

１．引言数值方法的动力特征近年引起了人们的广泛关注。其中之一就是系统的平衡态和数值方法的平衡态相一致的问题,即用一个数值方法沿定步长求解系统时,是否会出现伪平衡。不可能出现伪平衡的方法称为是正则的。ＲＫ方法和线性多步法的正则性已被众多的文献研究［２,３,４］,其它方法的正则性显然是一亟待研究的问题。本文讨论较ＲＫ方法和线性多步法远为广泛的一般线性方法的正则性。设ｆ：Ｒ~（Ｎ）→Ｒ~（Ｎ）是一充分光滑的映射,考虑求解初值问题：的一般线性方法［１］：其中步长逼近于逼近于关于微分方程真解ｙ（ｔ）在第ｎ层…     

由FR共轭梯度法控制的两类优化算法的全局收敛性

１引言 考虑无约束优化问题其中ｆ：Ｒｎ→Ｒ是一阶可微函数．求解（１）的非线性共轭梯度法具有如下形式：其中ｇｋ＝　ｆ（ｘｋ）,ａｋ是通过某种线搜索获得的步长,纯量βｋ的选取使得方法（２）—（３）在ｆ（ｘ）是严格凸二次函数且采用精确线搜索时化为线性共轭梯度法［１］．比较常见的βｋ的取法有Ｆｌｅｔｃｈｅｒ－Ｒｅｅｖｅｓ（ＦＲ）公式［２］和Ｐｏｌａｋ－Ｒｉｂｉｅｒｅ－Ｐｏｌｙａｋ（ＰＲＰ）公式［３－４］等．它们分别为其中　　　取欧几里得范数．对于一般非线性函数,ＦＲ方法具有较好的理论收敛性［５－６］,而…     

θ-方法的非线性渐近稳定性

１引言 数值求解延迟微分方程时,方法的稳定性具有无容置疑的重要性．自１９７５年Ｂａｒｗｅｌｌ引入Ｐ－稳定性与ＧＰ－稳定性概念以来,该领域研究已获许多重要成果（如［７］［８］）．它们大多是基于下面标量线性模型方程：其中λ,μ为复数且满足延迟量τ（＞０）为常数,函数θ（ｔ）连续． 我们首先回忆Ｂａｒｗｅｌｌ［１］的定义． 定义１．１一个数值方法称为是Ｐ－稳定的,如果对任意正整数ｒ用该方法按步长ｈ＝τ／ｒ求解（１．１）时在节点ｔｎ＝ｎｈ的数值解ｙｎ满足 定义１．２一个数值方法称为是ＧＰ－稳定的,如果用该方…     

解第一类边界积分方程的高精度机械求积法与外推

０．引言使用单层位势理论把Ｄｉｒｉｃｈｌｅｔ问题：转化为具有对数核的边界积分方程：这里Г假设为简单光滑闭曲线．熟知,若Г的容度Ｃｒ≠１,（０．２）有唯一解存在［１］．借助参数变换这里的数值解法有Ｇａｌｅｒｋｉｎ法［２］,配置法［３］,和谱方法~［４］,这些方法有一个共同缺点就是矩阵元素的生成要计算反常积分,由于离散方程的系数矩阵是满阵,使矩阵生成的工作量很庞大,甚至超过了解方程组的工作量．显然,如能找到适当求积公式离散（０．２）,则可节省大量计算．使用求积公式法解（０．２）的文献不多,［５］中提…     

W2^2（D）空间第一类算子方程近似解

０引言本文在具有再生核的Ｓｏｂｏｌｅｖ空间Ｗ＿２~２（Ｄ）中研究第一类算子方程的近似求解问题,本文利用再生核空间技巧得到的方程（１）的近似解ｕｎ（Ｍ）有如下特点：１近似解ｕｎ（Ｍ）的构成只需用到ｆ在有限个节点｛Ｗｉ｝１ｎ　Ｄ上的值;２当｛Ｍｉ｝１在Ｄ中稠密时,可以得到方程解析解ｕ（Ｍ）的表达式,并且而且误差随节点个数ｎ的增加接空间范数单调下降．数值算例表明,该方法是有效的．1Ｗ22（Ｄ）空间及其再生核记Ｄ=［ａ,ｂ］×［ｃ,ｄ］为有限区域,设是Ｄ上一组互异节点系．在［１］中,设Ｗ２２［ａ,ｂ］＝｛ｕ…     

关于一类色散型发展方程反问题的一个注记

关于一类色散型发展方程反问题的一个注记宋守根（中南工业大学地质系，长沙４１００８３）文献［１］利用Ｃ０半群理论研究一类非线性色散型方程的反问题．其中ｎ是Ｒ”中具有光滑边界ｏｆｆ的有界区域，面是ｎ维Ｌａｐｌａｃｅ算子，而算子Ｌ０为本文将改进［１］的一个...     

A cut-peak function method for global optimization

A new method is proposed for solving box constrained global optimization problems. The basic idea of the method is described as follows: Constructing a so-called cut-peak function and a choice function for each present minimizer, the original problem of finding a global solution is converted into an auxiliary minimization problem of finding local minimizers of the choice function, whose objective function values are smaller than the previous ones. For a local minimum solution of auxiliary problems this procedure is repeated until no new minimizer with a smaller objective function value could be found for the last minimizer. Construction of auxiliary problems and choice of parameters are relatively simple, so the algorithm is relatively easy to implement, and the results of the numerical tests are satisfactory compared to other methods.     

A Barrier Function Method for the Nonconvex Quadratic Programming Problem with Box Constraints

In this paper a barrier function method is proposed for approximating a solution of the nonconvex quadratic programming problem with box constraints. The method attempts to produce a solution of good quality by following a path as the barrier parameter decreases from a sufficiently large positive number. For a given value of the barrier parameter, the method searches for a minimum point of the barrier function in a descent direction, which has a desired property that the box constraints are always satisfied automatically if the step length is a number between zero and one. When all the diagonal entries of the objective function are negative, the method converges to at least a local minimum point of the problem if it yields a local minimum point of the barrier function for a sequence of decreasing values of the barrier parameter with zero limit. Numerical results show that the method always generates a global or near global minimum point as the barrier parameter decreases at a sufficiently slow pace.     

基于Spearman的rho的Copula参数模型的选择

从Spearman的rho与Kendall的tau的关系入手,讨论了一类二元Copula参数模型的选择问题.由于这类二元Copula参数模型的Spearman的rho与Kendall的tau存在某种函数关系,模型选择问题转化为了曲线拟合检验问题.对于正态Copula、Frank-Copula,FGM-Copula、B11-Copula等这类Copula参数模型,说明了两种情况下进行模型选择的方法,并对中国股市的上证指数与深证综指作了实证分析,结果表明两者存在着较强的正相关性,相关性模型选取B11-Copula参数模型最合适.     

一类具有连续变量的全局最优化问题的凸化方法

本文研究了一类非凸最优化问题的凸化方法与最优性条件的问题.利用构造含有参数的函数变换方法,将具有次正定性质的目标函数凸化,并获得了这一类非凸优化问题全局最优解的充要条件,推广了凸化方法在求解全局最优化问题方面的应用.     

Value-Estimation Function Method for Constrained Global Optimization

A novel value-estimation function method for global optimization problems with inequality constraints is proposed in this paper. The value-estimation function formulation is an auxiliary unconstrained optimization problem with a univariate parameter that represents an estimated optimal value of the objective function of the original optimization problem. A solution is optimal to the original problem if and only if it is also optimal to the auxiliary unconstrained optimization with the parameter set at the optimal objective value of the original problem, which turns out to be the unique root of a basic value-estimation function. A logarithmic-exponential value-estimation function formulation is further developed to acquire computational tractability and efficiency. The optimal objective value of the original problem as well as the optimal solution are sought iteratively by applying either a generalized Newton method or a bisection method to the logarithmic-exponential value-estimation function formulation. The convergence properties of the solution algorithms guarantee the identification of an approximate optimal solution of the original problem, up to any predetermined degree of accuracy, within a finite number of iterations.     

A variational-type method of fundamental solutions for a Cauchy problem of Laplace’s equation

In this paper, we propose a new regularization method based on a finite-dimensional subspace generated from fundamental solutions for solving a Cauchy problem of Laplace’s equation in a simply-connected bounded domain. Based on a global conditional stability for the Cauchy problem of Laplace’s equation, the convergence analysis is given under a suitable choice for a regularization parameter and an a-priori bound assumption to the solution. Numerical experiments are provided to support the analysis and to show the effectiveness of the proposed method from both accuracy and stability.     

一个新的单参数填充函数

填充函数法是求解全局优化问题的一种重要的确定性算法.本文将在前人的基础上,提出了一个新的单参数填充函数.并通过数值算例验证了该算法的有效性和可行性.     

A filled function method for finding a global minimizer on global integer optimization

The paper gives a definition of the filled function for nonlinear integer programming. This definition is modified from that of the global convexized filled function for continuous global optimization. A filled function with only one parameter which satisfies this definition is presented. We also discuss the properties of the proposed function and give a filled function method to solve the nonlinear integer programming problem. The implementation of the algorithm on several test problems is reported with satisfactory numerical results.     

Metrical characterization of conformal capacity zero

The introduction of a parameter in the fixed point equation yields a family of operators with identical fixed points. A method is presented for minimization of the Lipschitz constant as a function of this parameter. This yields an operator with an optimal error estimate. The choice of a suitable weighting function is essential. As an example, a nonlinear boundary value problem is treated. A one-sided Lipschitz condition is sufficient.     

A discrete dynamic convexized method for nonlinear integer programming

In this paper, we consider the box constrained nonlinear integer programming problem. We present an auxiliary function, which has the same discrete global minimizers as the problem. The minimization of the function using a discrete local search method can escape successfully from previously converged discrete local minimizers by taking increasing values of a parameter. We propose an algorithm to find a global minimizer of the box constrained nonlinear integer programming problem. The algorithm minimizes the auxiliary function from random initial points. We prove that the algorithm can converge asymptotically with probability one. Numerical experiments on a set of test problems show that the algorithm is efficient and robust.