超临界系统自发启动概率计算

引言 我们曾求解过死绝方程,这种计算问题的核心是用Monte Carlo方法解非定常中子输运方程,同时还要解高次代数方程组。由于用Monte Carlo方法解非定常中子输运问题已有了介绍(见[1],[2]及[3]),于是本文重点在于求高次代数方程组的解,对     

具有初始缺陷弹性板的稳定性分析

本文以Marguerre方程为基础,用奇异性理论研究了初始挠度缺陷以及横向载荷对弹性板屈曲后分叉解的影响。借助于普适开折的原理,在单特征值局部邻域内将该问题的失稳分析转化为三次代数方程的讨论,从而确定出分叉解的性态。同时绘出了在不同参数下的分叉解文,讨论了几何缺陷和横向载荷对特征值的影响。     

点时滞系统的反馈镇定

求解特征矩阵是镇定时滞系统的关键问题，本文给出了系统的特征根的代数重复度与几何重复度均为一般值的情况下特征矩阵的求法，即把它归结为求一组线性代数方程的问题，并得到了该方程组有组及对应于同一特征值的解向量组线性独立的充分条件。本文还提出了一种算法来处理系统对应于不同特征值的左行征向量线性相关情况下系统的镇定问题最后，举例说明了设计步骤。     

分叉问题的几何描述及其计算方法

本文通过一个构造的向量场,建立了非线性代数方程组的解流形与对应向量场的不变流形之间的等价关系,把解流形的分叉问题转化为对向量场奇点附近局部性态的研究。这种思路具有几何直观性,并为解流形的跟踪与分叉计算提供了一套新的数值方法。     

正规升列在参数代数方程组求解中的应用

本文提出一种利用多项式系统的正规零点分解的算法来求解代数方程组以及带有参数的代数方程组的方法．对于给定的的代数方城组,通过正规分解,可以得到一组具有三角形式的分解．根据这种三角形式,我们可以给出代数方程组的所有解．而对于带有参数方程组,将给出方程组有解时参数需满足的条件．进一步,对于给定的参数值,正规分解中得到三角形式仍然保持,通过求解三角形式的方程组从而得出原参数方程组的解．     

计算代数方程组孤立奇异解的符号数值方法

求解代数方程组是计算代数几何的最基本问题之一,孤立奇异解的计算则是其中最具挑战性的课题之一,在科学与工程计算中有着广泛的应用,如机器人、计算机视觉、机器学习、人工智能、运筹学、密码学和控制论等.本文结合作者的研究成果,综述了符号数值方法在计算代数系统孤立奇异解、特别是近似奇异解精化与验证方面的研究进展,并对未来的研究方向提出了展望.     

解綫代数方程组的迭代法及其迭代过程的收斂条件

前言本文的目的,在簡要介紹解綫代数方程組的迭代法之后,主要是对于这种迭代过程的收斂条件問題,从級数收斂方面作了一些考虑,避开了通常方法所涉及到的有关矩陣的特征值和特征向量、向量和矩陣的范数以及矩陣的相似变換等一系列的綫代数理論,而仅仅用到較少的数学分析知識,同样給出了通常的两个收斂性定理及收斂速度估計式。它在教学上提供了一个可以采用的处理教材的方法,也为具有初等分析知識的数学工作者全面掌握这一方法探索到了一个簡便途径。§1.解綫代数方程組的迭代法 設給定n阶綫代数方程組为     

微分求积法分析平面接头应力奇异性北大核心CSCD

对于双材料平面接头问题提出了一个分析应力奇性指数的新方法:微分求积法(DQM).首先,将平面接头连接点处位移场的径向渐近展开格式代入平面弹性力学控制方程,获得了关于应力奇性指数的常微分方程组(ODEs)特征值问题.然后,基于DQM理论,将ODEs的特征值问题转化为标准型广义代数方程组特征值问题,求解之可一次性地计算出双材料平面接头连接点处应力奇性指数,同时,一并求出了接头连接点处相应的位移和应力特征函数.数值计算结果说明该文DQM计算平面接头连接点处应力奇性指数的结果是正确的.     

连续型线性定常系统的Riccati代数方程的小摄动问题

本文讨论了连续型线性定常系统摄动的Riccati代数方程所对应的稳定性问题.通过矩阵范数分析建立了摄动的Riccati代数方程的解的摄动界估计(以系统参数摄动界表出),从而提供了一种方便的实用计算方法.     

Zakharov—Shabat反散射方程的不可约形式

对求孤子解的Z-S反散射方程组,给出了它的等价的不可约形式。在求N-孤子解时,Z-S方程组由2N个线性代数方程组成,求解手续就是计算一个2N×2N矩阵之逆。而本文所得的它的不可约形式,是由N个线性代数方程组成的,求解只需计算一个N×N矩阵之逆,从而使计算量大大缩小。文未给出求两孤子解和呼吸子解的实际的简单计算和结果。     

Lanczos-type Methods for Continuation Problems

We study the Lanczos type methods for continuation problems. First we indicate how the symmetric Lanczos method may be used to solve both positive definite and indefinite linear systems. Furthermore, it can be used to monitor the simple bifurcation points on the solution curve of the eigenvalue problems. This includes computing the minimum eigenvalue, the minimum singular value, and the condition number of the partial tridiagonalizations of the coefficient matrices. The Ritz vector thus obtained can be applied to compute the tangent vector at the bifurcation point for branch-switching. Next, we indicate that the block or band Lanczos method can be used to monitor the multiple bifurcations as well as to solve the multiple right hand sides. We also show that the unsymmetric Lanczos method can be exploited to compute the minimum eigenvalue of a nearly symmetric matrix, and therefore to detect the simple bifurcation point as well. Some preconditioning techniques are discussed. Sample numerical results are reported. Our test problems include second order semilinear elliptic eigenvalue problems. © 1997 by John Wiley & Sons, Ltd.     

Improving directions of negative curvature in an efficient manner

In order to converge to second-order KKT points, second derivative information has to be taken into account. Therefore, methods for minimization satisfying convergence to second-order KKT points must, at least implicitly, compute a direction of negative curvature of an indefinite matrix. An important issue is to determine the quality of the negative curvature direction. This problem is closely related to the symmetric eigenvalue problem. More specifically we want to develop algorithms that improve directions of negative curvature with relatively little effort, extending the proposals by Boman and Murray. This paper presents some technical improvements with respect to their work. In particular, we study how to compute “good” directions of negative curvature. In this regard, we propose a new method and we present numerical experiments that illustrate its practical efficiency compared to other proposals.     

Accurate solutions of fourth order Sturm-Liouville problems

Recently we introduced a new method which we call the Extended Sampling Method to compute the eigenvalues of second order Sturm-Liouville problems with eigenvalue dependent potential. We shall see in this paper how we use this method to compute the eigenvalues of fourth order Sturm-Liouville problems and present its practical use on a few examples.     

Verified numerical computations for multiple and nearly multiple eigenvalues of elliptic operators

In this paper, we propose a numerical method to verify bounds for multiple eigenvalues for elliptic eigenvalue problems. We calculate error bounds for approximations of multiple eigenvalues and base functions of the corresponding invariant subspaces. For matrix eigenvalue problems, Rump (Linear Algebra Appl. 324 (2001) 209) recently proposed a validated numerical method to compute multiple eigenvalues. In this paper, we extend his formulation to elliptic eigenvalue problems, combining it with a method developed by one of the authors (Jpn. J. Indust. Appl. Math. 16 (1998) 307).     

The sampling method for sturm-liouville problems with the eigenvalue parameter in the boundary condition.

ABSTRACT. We shall apply the sampling method to compute the eigenvalues of a regular Sturm Liouville problem when a boundary condition contains the eigenvalue parameter. Truncation errors will be worked out and help us obtain eigenvalue enclosures. Examples are provided to illustrate the theory.     

Preconditioned iterative methods for a class of nonlinear eigenvalue problems

This paper proposes new iterative methods for the efficient computation of the smallest eigenvalue of symmetric nonlinear matrix eigenvalue problems of large order with a monotone dependence on the spectral parameter. Monotone nonlinear eigenvalue problems for differential equations have important applications in mechanics and physics. The discretization of these eigenvalue problems leads to nonlinear eigenvalue problems with very large sparse ill-conditioned matrices monotonically depending on the spectral parameter. To compute the smallest eigenvalue of large-scale matrix nonlinear eigenvalue problems, we suggest preconditioned iterative methods: preconditioned simple iteration method, preconditioned steepest descent method, and preconditioned conjugate gradient method. These methods use only matrix-vector multiplications, preconditioner-vector multiplications, linear operations with vectors, and inner products of vectors. We investigate the convergence and derive grid-independent error estimates for these methods. Numerical experiments demonstrate the practical effectiveness of the proposed methods for a model problem.     

A Fast Space-decomposition Scheme for Nonconvex Eigenvalue Optimization

In this paper, a nonsmooth bundle algorithm to minimize the maximum eigenvalue function of a nonconvex smooth function is presented. The bundle method uses an oracle to compute separately the function and subgradient information for a convex function, and the function and derivative values for the smooth mapping. Using this information, in each iteration, we replace the smooth inner mapping by its Taylor-series linearization around the current serious step. To solve the convex approximate eigenvalue problem with affine mapping faster, we adopt the second-order bundle method based on ????-decomposition theory. Through the backtracking test, we can make a better approximation for the objective function. Quadratic convergence of our special bundle method is given, under some additional assumptions. Then we apply our method to some particular instance of nonconvex eigenvalue optimization, specifically: bilinear matrix inequality problems.     

Computing the lower and upper bounds of Laplace eigenvalue problem: by combining conforming and nonconforming finite element methods

We introduce some ways to compute the lower and upper bounds of the Laplace eigenvalue problem.By using the special nonconforming finite elements,i.e.,enriched Crouzeix-Raviart element and extended Q1ro t,we get the lower bound of the eigenvalue.Additionally,we use conforming finite elements to do the postprocessing to get the upper bound of the eigenvalue,which only needs to solve the corresponding source problems and a small eigenvalue problem if higher order postprocessing method is implemented.Thus,we can obtain the lower and upper bounds of the eigenvalues simultaneously by solving eigenvalue problem only once.Some numerical results are also presented to demonstrate our theoretical analysis.     

一类张量特征值互补问题

矩阵特征值互补问题在力学系统领域有广泛的应用.在本文中,我们提出了一类特殊的四阶张量特征值互补问题,它是矩阵特征值互补问题的推广.我们对该特征值互补问题解的存在性,计算复杂度等性质进行了初步的研究.在一定条件下,我们建立了该互补问题同一类非线性约束优化问题的等价性联系,并由此提出了平移投影幂法来求解该特征值互补问题.     

Linearization of multipolynomials and applications

We prove that every multipolynomial between Banach spaces is the composition of a canonical multipolynomial with a linear operator, and that this correspondence establishes an isometric isomorphism between the spaces of multipolynomials and linear operators. Applications to composition ideals of multipolynomials and to multipolynomials that are of finite rank, approximable, compact, and weakly compact are provided.