Some explicit formulas for mixed exponential service systems

We present some explicit formulas for queue length and waiting time distributions of customers in the M/HE_m/1 queue. The formulas are obtained with the aid of roots of quadratic, cubic, and quartic polynomials constructed from a recurrence equation. With an example, we demonstrate that the formulas for queueing distributions are extremely accurate, while the corresponding infinite history M/GI/1 recurrence equation is not. Applications include computation of queueing distributions, accurate tail probabilities, and in systems where exponentiality can be replaced by hyperexponentiality. The explicit solutions are easier to use than the problem-specific partial fraction expansions of the Pollachek-Khinchin transform.     

Some computational aspects of the matrix exponential

Matrix norms that satisfy the Schwarz inequality are used to determine upper bounds for the error in some common computations involving the matrix exponential function.     

A self-validating numerical method for the matrix exponential

An algorithm is presented, which produces highly accurate and automatically verified bounds for the matrix exponential function. Our computational approach involves iterative defect correction, interval analysis and advanced computer arithmetic. The algorithm presented is based on the “scaling and squaring” scheme, utilizing Padé approximations and safe error monitoring. A PASCAL-SC program is reported and numerical results are discussed.     

Computing integrals involving the matrix exponential

A new algorithm for computing integrals involving the matrix exponential is given. The method employs diagonal Padé approximation with scaling and squaring. Rigorous truncation error bounds are given and incorporated in a Fortran subroutine. The computational aspects of this program are discussed and compared with existing techniques.     

可分型指数矩阵的快速精细积分法

针对可分型矩阵的特性,结合2N类算法为可分型指数矩阵的计算提出一种快速精细积分法.核心思想是:利用可分型矩阵中的子矩阵进行分块计算;增加Taylor展开式的保留项数,减少迭代次数.一方面,程序实现简便,另一方面,数值算例表明:对矩阵维数很大的可分型指数矩阵计算来说,本文的快速精细积分法减少了计算量和存储量,大大地提高了计算效率.     

h-space structure in matrix displacement formulas

A class of spaces of matrices, calledh-spaces, is considered, extending previous results in [R. Bevilacqua, P. Zellini,Closure, commutativity and minmal complexity of some space of matrices, Linear and Multilinear Algebra,25, (1989) 1–25]. These spaces include several known classes of matrix algebras, such as group matrix algebras and Hessenberg algebras and, in particular, certain symmetric closed 1-spaces, which are structurally related to Toeplitz plus Hankel-like matrices. Following the displacement rank technique, these spaces are involved in general displacement decomposition formulas of an arbitrary matrixA. These decompositions lead to a significant representation formula for the inverse of a centrosymmetric Toeplitz plus Hankel matrix.     

Some integration formulas for a four-node isoparametric element

In this paper we present analytical integration formulas for a four-node isoparametric element. All the formulas are constructed with three simple functions of element nodal values, and therefore they can be easily coded and have high cost-effectiveness. Some of the formulas are exact and some are approximate. However, even the approximate formulas have enough accuracy and do not suffer from the so-called “hourglass” instability, which is a major drawback of one-point quadrature.     

Error analysis of two algorithms for the computation of the matrix exponential

We analyze two algorithms for the computation of the matrix exponential: the Taylor Series and the Scaling and Squaring methods. We give new upper bounds on the roundoff and truncation errors introduced, and we present some numerical experiments, comparing the actual outcomes to the theoretical error bounds. We analyze, in detail, the reasons why the Taylor Series method can compete with Scaling and Squaring method if the norm of the original matrix is less than one.     

Fourier exponential series matrix of integration

The Fourier exponential operational matrix of integration P is derived which is analogous to that previously derived for other types of orthogonal functions. This matrix P may be used to solve problems such as identification, analysis and optimal control.     

Generating correlated matrix exponential random variables

In this paper, we focus on inter-arrival time autocorrelation and its impact on model performance. We present a technique to generate matrix exponential random variables that match first-order statistics (moments) and second-order statistics (autocorrelation) from an empirical distribution. We briefly explain the matrix exponential distribution and show that we can represent any empirical distribution arbitrarily closely as matrix exponential. We then show how we can incorporate an autocorrelation structure into our matrix exponential random variables using the autoregressive to anything technique. We present examples showing how we match first and second-order statistics from empirical distributions and finally we show that our autocorrelation matrix exponential random variables produce more accurate performance metrics from simulation models than traditional techniques.     

Accurate and efficient matrix exponential computation

This work gives a new formula for the forward relative error of matrix exponential Taylor approximation and proposes new bounds for it depending on the matrix size and the Taylor approximation order, providing a new efficient scaling and squaring Taylor algorithm for the matrix exponential. A Matlab version of the new algorithm is provided and compared with Padé state-of-the-art algorithms obtaining higher accuracy in the majority of tests at similar or even lower cost.     

On deriving explicit formulas of the surface areas for the arrangement graphs and some of the related graphs

We derive an explicit formula for the surface area of the arrangement graph, i.e. the number of vertices at a certain distance from the identity vertex in such a graph. We also present such formulas for the star graph, the alternating group graph, and the split-star graph, via their respective structural relationship to the arrangement graph.     

Some properties of log-convex function and applications for the exponential function

In this paper, some properties of log-convex function are researched, and integral inequalities of log-convex functions are proved. As an application, an estimation formula of remainder terms in Taylor series expansion is given.     

Some generalised results on exponential delays

Exponential delays are widely used in literatures on system dynamics due primarily to its simplicity of form and to its adequacy to fit our usual degree of knowledge about the actual systems to be represented. An attempt has been made in this paper to arrive at certain useful results through two generalised methods for the analysis o f transient of these delays to changes in their inflow rate.     

基于显式与隐式反馈信息的概率矩阵分解推荐

针对现有的基于用户显式反馈信息的推荐系统推荐准确率不高的问题,提出了一种基于显式与隐式反馈信息的概率矩阵分解推荐方法。该方法综合考虑了显示反馈信息和隐式反馈信息,在对用户信任关系矩阵和商品评分矩阵进行概率分解的同时加入了用户评分记录的隐式反馈信息,优化训练模型参数,为用户提供精确的预测评分。实验结果表明,该方法可以有效地获得用户偏好,产生大量的准确度高的推荐。     

Some representations for the Drazin inverse of a modified matrix

In this paper, we give some results for the Drazin inverse of a modified matrix \(M=A-CD^dB\) with the generalized Schur complement \(Z=D-BA^dC\) under some conditions. Further, we present some new results for the Drazin inverse of the modified matrix \(M=A-CD^dB\) , when the generalized Schur complement \(Z=0\) under some conditions. As a result, some conclusions are obtained directly from our results.     

Necessary conditions for the exponential stability of time‐delay systems via the Lyapunov delay matrix

Exponential necessary stability conditions for linear systems with multiple delays are presented. The originality of these conditions is that, in analogy with the case of delay free systems, they depend on the Lyapunov matrix function of the delay system. They are validated by examples for which the analytic characterization of the stability region is known. Copyright © 2013 John Wiley & Sons, Ltd.     

Time discretization of nonlinear time-delay system using matrix exponential method

A time discretization method for nonlinear time-delay systems is proposed in this paper. The proposed method is based on the matrix exponential method and includes the automatic correction of rounding errors. It is robust to ill-conditioned problems and suitable for any nonlinear system. In the proposed algorithm, each sampling time interval is divided into two subintervals to be considered separately according to the time delay and the sampling period. The performance of the proposed discretization procedure is evaluated by two case studies.     

对角矩阵指数优化的局部保持映射算法

局部保持映射（LPP）算法利用欧几里德距离求得权值累加得到对角矩阵,利用结果进行降维。对于这个算法是否可以进一步优化还值得进一步探讨。对该算法所依据的公式进行修改,在对角矩阵上引入指数参数,形成对角距阵指数优化的局部保持映射算法。通过实验可以证明,对角距阵指数优化的局部保持映射算法能够影响降维的结果,可以使得降维更容易得到接近本征维数的投影向量,通过实验验证降维后的识别效果和对噪声的敏感度。