基于亏基的摄动原始单纯形Ⅰ阶段算法

通过摄动技术来使问题强制获得对偶可行性,执行亏基对偶单纯形算法得到一个原始可行基,并采用修正的主元规则,以充分发挥这两种算法的优势,从而为亏基原始单纯形算法提供一个新的I阶段算法,以使其进一步克服退化所带来的困扰.初步的数值试验表明,亏基和摄动两种算法优势的结合,能有效地克服退化的影响,能有效地减少总迭代次数和运行时间,其效率远远优于传统两阶段单纯形算法.     

单纯形法选择进出基变元的一个新准则

解线性规划单纯形法迭代中,G.B.Dantzig等人给出的进基原则看似简单,但其忽略了影响目标函数增加量的另外一个因素—进基变元的产出系数,而试图给出一个新的迭代进出基准则—最大增量准则,一方面可以加快迭代速度,同时也可以避免迭代中可能遇到的所谓循环.     

无比值检验的亏基原始Ⅰ阶段算法

首次将亏基和无比值检验列主元规则相结合,执行亏基对偶单纯形算法得到一个原始可行基,以充分发挥这两种算法的优势,从而为亏基原始单纯形算法提供一个新的I阶段算法,以使其进一步克服退化所带来的困扰.数值试验表明,亏基和无比值主元规则的结合,能有效地减少总迭代次数和运行时间,其效率远远优于传统两阶段单纯形算法.     

单形构造定理的一种代数证法

给出n维欧氏空间En中n维单形构造定理的一种代数证法,这种代数证法十分简洁.     

On Iterations of the Green Integrals and their Applications to Elliptic Differential Complexes

Convergence of iterations of special Green integrals for overdetermined elliptic linear partial differential operators P of order p ≥ 1 is proved. Using this result we obtain necessary and sufficient conditions for the solvability of the equation Pu = f in the Sobolev space W^p,²(D) and, as a corollary, necessary and sufficient conditions for the vanishing of the first cohomology group of elliptic differential complexes. Also a criterion for the solvability of a P-Neumann problem for elliptic differential operators is proved.     

Backtracking Adaptive Search: Distribution of Number of Iterations to Convergence

Backtracking adaptive search is a simplified stochastic optimiza-tion procedure which permits the acceptance of worsening objective function values. It generalizes the hesitant adaptive search, which in turn is a gener-alization of the pure adaptive search. In this paper, we use ideas from the theory of stochastic processes to determine the full distribution of the number of iterations to convergence for the backtracking adaptive search.Communicated by P. M. PardalosThe authors thank theMarsden Fund of the Royal Society of New Zealand for support of this research. The fourth author was supported by a Bright Future Scholarship administered by the Foundation for Research, Science and Technology.     

Distortion Growth for Iterations of Diffeomorphisms of the Interval

We obtain several results on the distortion asymptotics for the iterations of diffeomorphisms of the interval extending the recent work of Polterovich and Sodin.     

如何利用单纯形表上的信息

本文分析了求解线性规划的基本方法－－单纯形法所使用的单纯形表，将表中所提供的信息分为直接信息和间接信息两类，论述了如何充分利用这些信息的方法。例如如何由最终表求原问题、如何利用表中的数据互相推演和校正等。这是一篇教学经验的总结，对初学者可能有一定的帮助。     

具误差的三种迭代收敛的等价性

考虑了具误差的Mann迭代,Ishikawa迭代和三重迭代对中间意义下的渐进非扩张映射和强逐次伪压缩映射收敛的等价性.我们的主要结果改善和推广了近期该方向研究所得到的某些成果.     

SOR和AOR迭代收敛的新判别准则

充分条件1和2,对于SOR迭代(0<ω≤1)和AOR迭代(0≤r≤1,0<ω≤1)也是适用的。那么,充分条件3对于SOR迭代和AOR迭代是否适用呢?迄今为止尚没有讨论过。这里我们给予肯定的回答。我们的结论基于如下两个引理。 引理1 如果A的主对角元全不为零,且满足条件(4),则detA≠0。 引理2 对线性方程组(1)的迭代法     

On the sensitivity of the LU factorization

This paper gives sensitivity analyses by two approaches forL andU in the factorizationA=LU for general perturbations inA which are sufficiently small in norm. By the matrix-vector equation approach, we derive the condition numbers for theL andU factors. By the matrix equation approach we derive corresponding condition estimates. We show how partial pivoting and complete pivoting affect the sensitivity of the LU factorization. The material presented here is a part of the first author's PhD thesis under the supervision of the second author. This research was supported by NSERC of Canada Grant OGP0009236.     

On the Convergence Theory of Preconditioned Subspace Iterations for Eigenvalue Problems

We consider preconditioned subspace iterations for the numerical solution of discretized elliptic eigenvalue problems. For these iterative solvers, the convergence theory is still an incomplete puzzle. We generalize some results from the classical convergence theory of inverse subspace iterations, as given by Parlett, and some recent results on the convergence of preconditioned vector iterations. To this end, we use a geometric cone representation and prove some new trigonometric inequalities for subspace angles and canonical angles. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A generalization of the implicit LU algorithm to an arbitrary initial matrix

The implicit LU algorithm of the (basic) ABS class corresponds to the parameter choices H ₁=I, z _i =w _i =e _i . The algorithm can be considered as the ABS version of the classic LU factorization algorithm. In this paper we consider the generalization where the initial matrix H₁ is arbitrary except for a certain condition. We prove that every algorithm in the ABS class is equivalent, in the sense of generating the same set of search directions, to a generalized implicit LU algorithm, with suitable initial matrix, that can be interpreted as a right preconditioning matrix. We discuss some consequences of this result, including a straightforward derivation of Bienaymé's (1853) classical result on the equivalence of the Gram–Schmidt orthogonalization procedure with Gaussian elimination on the normal equations. This revised version was published online in June 2006 with corrections to the Cover Date.     

Iterations of anti-selfdual Lagrangians and applications to Hamiltonian systems and multiparameter gradient flows

Anti-selfdual Lagrangians on a state space lift to path space provided one adds a suitable selfdual boundary Lagrangian. This process can be iterated by considering the path space as a new state space for the newly obtained anti-selfdual Lagrangian. We give here two applications for these remarkable permanence properties. In the first, we establish for certain convex–concave Hamiltonians on a–possibly infinite dimensional–symplectic space H ², the existence of a solution for the Hamiltonian system that connects in a given time T > 0, two Lagrangian submanifolds. Another application deals with the construction of multiparameter flows, including those generated by vector fields that represent superpositions of skew-adjoint operators with gradients of convex potentials. Our methods are based on the new variational calculus for anti-selfdual Lagrangians developed in [5–7].     

单纯形法检验数的新计算方法

单纯形法仍然是求解线性规划最具竞争力的算法之一,改进它的计算效率仍具有理论和现实意义.本文通过改进检验数的计算方式,提出了一种实施单纯形法新的计算方式.这种计算方式方便简单,无论采用单纯形表还是采用数值迭代计算都可以提高计算效率.     

On the forms of the truncated LU decomposition

Matrix LU decomposition has six ijk forms. Different forms have different computational complexities and storage requirements, particularly on vector and parallel computers. Other factors governing the choice of a particular form are considered. For treating Fredholm integral equations of the first kind, the truncated LU decomposition of the resulting system matrix is recommended. Required modifications to selected known ijk forms are presented.