关于矩阵分解为稳定矩阵之和的讨论

要解决的问题是一个矩阵是否可以分解为若干稳定 (连续时间意义下 )矩阵的和 .通过推广Ito,Hattori and Maeda在文献 [2 ]中使用的方法并运用其中的成果 ,我们得到了更为准确的相关结论 .     

On the distributions of some test criteria for a covariance matrix under local alternatives and bootstrap approximations

The asymptotic distribution of some test criteria for a covariance matrix are derived under local alternatives. Except for the existence of some higher moments, no assumption as to the form of the distribution function is made. As an illustration, a case of t distribution included normal model is considered and the power of the likelihood ratio test and Nagao's test for sphericity, as described in Srivastava and Khatri and Anderson, is computed. Also, the power is computed using the bootstrap method. In the case of t distribution, the bootstrap approximation does not appear to be as good as the one obtained by the asymptotic expansion method.     

Roundoff error analysis of fast DCT algorithms in fixed point arithmetic

Discrete cosine transforms (DCT) are essential tools in numerical analysis and digital signal processing. Processors in digital signal processing often use fixed point arithmetic. In this paper, we consider the numerical stability of fast DCT algorithms in fixed point arithmetic. The fast DCT algorithms are based on known factorizations of the corresponding cosine matrices into products of sparse, orthogonal matrices of simple structure. These algorithms are completely recursive, are easy to implement and use only permutations, scaling, butterfly operations, and plane rotations/rotation-reflections. In comparison with other fast DCT algorithms, these algorithms have low arithmetic costs. Using von Neumann–Goldstine’s model of fixed point arithmetic, we present a detailed roundoff error analysis for fast DCT algorithms in fixed point arithmetic. Numerical tests demonstrate the performance of our results.      

Stability and error analysis on partially implicit schemes

Subdomain techniques have been widely used for solving elliptic and parabolic equations. For parabolic problems, it is possible to combine subdomain techniques with explicit methods to construct efficient algorithms. In addition, this kind of algorithms is naturally suitable for parallel computing. However, the stability of such schemes has been considered as a very difficult issue. In this article, we use an exact error propagation and discrete scheme smoothing approach to give a posteriori stability and error analysis. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

二元齐次矩阵Pad6．型逼近及误差公式

二元矩阵Pad6一型逼近的计算比较复杂．本文受Benouahmane和Cuyt的启发,通过引入一种变量代换,将二元齐次矩阵形式幂级数转化为一元含参数形式的矩阵形式幂级数,并给出了二元齐次矩阵Pad6一型逼近的构造性的定义和误差公式的证明．数值实例说明了此方法的有效性．     

An adaptive improvement on PageRank algorithm

In this article, we introduce the Google’s method for quality ranking of web page in a formal mathematical format, use the power iteration to improve the PageRank, and also discuss the effect of different q to the PageRank, as well as how a PageRank will be changed if more links are added to one page or removed from some pages.     

A modified algorithm for the Perron root of a nonnegative matrix

An algorithm of diagonal transformation for the Perron root of nonnegative matrices is proposed by Duan and Zhang [F. Duan, K. Zhang, An algorithm of diagonal transformation for Perron root of nonnegative irreducible matrices, Appl. Math. Comput. 175 (2006) 762-772]. This method can be used for all nonnegative irreducible matrices. In this paper, an improved algorithm which is based on this method is proposed. The new algorithm inherits all the above-mentioned advantages of the original algorithm and has higher efficiency. It is testified by numerical testing that the efficiency of the new algorithm is improved greatly.     

A remark on the GOP algorithm for global optimization

In this paper it is shown that a large class of smooth mathematical programming problems can be converted into the standard forms to which the GOP algorithm applies.     

Unconditionally Stable Pressure-Correction Schemes for a Linear Fluid-Structure Interaction Problem

We consider in this paper numerical approximation of the linear Fluid-Structure Interaction (FSI). We construct a new class of pressure-correction schemes for the linear FSI problem with a fixed interface, and prove rigorously that they are unconditionally stable. These schemes are computationally very efficient, as they lead to, at each time step, a coupled linear elliptic system for the velocity and displacement in the whole region and a discrete Poisson equation in the fluid region.     

Stable iterations for the matrix square root

Any matrix with no nonpositive real eigenvalues has a unique square root for which every eigenvalue lies in the open right half-plane. A link between the matrix sign function and this square root is exploited to derive both old and new iterations for the square root from iterations for the sign function. One new iteration is a quadratically convergent Schulz iteration based entirely on matrix multiplication; it converges only locally, but can be used to compute the square root of any nonsingular M-matrix. A new Padé iteration well suited to parallel implementation is also derived and its properties explained. Iterative methods for the matrix square root are notorious for suffering from numerical instability. It is shown that apparently innocuous algorithmic modifications to the Padé iteration can lead to instability, and a perturbation analysis is given to provide some explanation. Numerical experiments are included and advice is offered on the choice of iterative method for computing the matrix square root.     

Backward error analysis of specified eigenpairs for sparse matrix polynomials

This article studies the unstructured and structured backward error analysis of specified eigenpairs for matrix polynomials. The structures we discuss include $T$-symmetric, $T$-skew-symmetric, Hermitian, skew Hermitian, $T$-even, $T$-odd, $H$-even, $H$-odd, $T$-palindromic, $T$-anti-palindromic, $H$-palindromic, and $H$-anti-palindromic matrix polynomials. Minimally structured perturbations are constructed with respect to Frobenius norm such that specified eigenpairs become exact eigenpairs of an appropriately perturbed matrix polynomial that also preserves sparsity. Further, we have used our results to solve various quadratic inverse eigenvalue problems that arise from real-life applications.     

Matrix analysis for associated consistency in cooperative game theory

Hamiache axiomatized the Shapley value as the unique solution verifying the inessential game property, continuity and associated consistency. Driessen extended Hamiache’s axiomatization to the enlarged class of efficient, symmetric, and linear values. In this paper, we introduce the notion of row (resp. column)-coalitional matrix in the framework of cooperative game theory. The Shapley value as well as the associated game are represented algebraically by their coalitional matrices called the Shapley standard matrix M^Sh and the associated transformation matrix M_λ, respectively. We develop a matrix approach for Hamiache’s axiomatization of the Shapley value. The associated consistency for the Shapley value is formulated as the matrix equality M^Sh = M^Sh · M_λ. The diagonalization procedure of M_λ and the inessential property for coalitional matrices are fundamental tools to prove the convergence of the sequence of repeated associated games as well as its limit game to be inessential. In addition, a similar matrix approach is applicable to study Driessen’s axiomatization of a certain class of linear values. In summary, it is illustrated that matrix analysis is a new and powerful technique for research in the field of cooperative game theory.     

关于矩阵的Bergstrom型不等式的修正

本文指出了[1]中关于实数域上亚正定阵和[3]中关于四元数体上正定自共轭矩矩阵的Bergstrom型不等式推广的错误,同时给出有关结论的正确形式.     

Homogeneous Models and Stable Diagrams

Some fragment is studied of stability theory in the category of D-sets. Conditions are given for existence of D-homogeneous models of however large power. A categoricity theorem is proven for the class of (D,)-homogeneous models.     

一类稳定的连续时间广义预测控制

引入无穷时域的1-范数性能指标,通过施加新的终端等式约束确定出无穷时域性能指标的一个上界,将不可解的优化问题转化为可解的优化问题,从而提出保证连续时间广义预测控制闭环稳定性的准无穷时域方法.仿真例子证明了算法的有效性.     

广义周期七对角矩阵的求逆新算法

讨论了广义周期七对角矩阵的求逆问题,利用七对角矩阵的特殊结构,通过矩阵的广义LU分解,给出了一种求解广义周期七对角逆矩阵的新型算法,该算法不需要对矩阵的各阶顺序主子式做任何限制并且适用于多种计算机代数系统,如：Mathematics,Macsyma,Matlab和Maple等.最后通过算例来说明了算法的有效性。     

Finite difference method and its convergent error analyses for thermistor problem

The thermistor problem is an initial-boundary value problem of coupled nonlinear differential equations. The nonlinear PDEs consist of a heat equation with the Joule heating as asource and a current conservation equation with temperature-dopendent electrical conductivity.This problem has important opplicatioJls in industry. In this paper, A new finite differencescheme is proposed on nonuniform rectangular partition for the thermistor problem. In the theo-retical analyses,the second-order error estimates are obtained for electrical potential in discrete L^2 and H^1 norms,and for the temperature in L^2 norm. In order to get these second-order errorestimates,the Joule heating source is used in a changed equivalent form.     

An error analysis for the calculation of semiintegrals and semiderivatives by the RL algorithm

The numerical approximation of the semiintegral and semiderivative of a function f by the RL algorithm of Oldham and Spanier [1] is examined. An error analysis is given for the case when f has a continuous second derivative. The performance of the algorithm when applied to experimental data is also discussed. Numerical examples are presented to illustrate the theory.     

Convergence analysis and error estimates for a parallel algorithm for solving the Navier-Stokes equations

Summary. This paper is concerned with the analysis of the convergence and the derivation of error estimates for a parallel algorithm which is used to solve the incompressible Navier-Stokes equations. As usual, the main idea is to split the main differential operator; this allows to consider independently the two main difficulties, namely nonlinearity and incompressibility. The results justify the observed accuracy of related numerical results. Received April 20, 2001 / Revised version received May 21, 2001 / Published online March 8, 2002 RID="*" ID="*" Partially supported by D.G.E.S. (Spain), Proyecto PB98–1134 RID="**" ID="**" Partially supported by D.G.E.S. (Spain), Proyecto PB96–0986 RID="**" ID="**" Partially supported by D.G.E.S. (Spain), Proyecto PB96–0986 RID="*" ID="*" Partially supported by D.G.E.S. (Spain), Proyecto PB98–1134 RID="**" ID="**" Partially supported by D.G.E.S. (Spain), Proyecto PB96–0986 RID="**" ID="**" Partially supported by D.G.E.S. (Spain) Proyecto PB96–0986     

计算Hamilton矩阵特征值的一个稳定的有效的保结构的算法

提出了一个稳定的有效的保结构的计算Hamilton矩阵特征值和特征不变子空间的算法,该算法是由SR算法改进变形而得到的。在该算法中,提出了两个策略,一个叫做消失稳策略,另一个称为预处理技术。在消失稳策略中,通过求解减比方程和回溯彻底克服了Bunser Gerstner和Mehrmann提出的SR算法的严重失稳和中断现象的发生,两种策略的实施的代价都非常低。数值算例展示了该算法比其它求解Hamilton矩阵特征问题的算法更有效和可靠。