Modelling Sales

This paper is concerned with some aspects of direct and indirect sales rates of products, and with a model based on the sales matrix. The purpose of this consideration is to determine, through indirect sales rates, the structure of sales and to construct a model which may be used for planning and forecasting goals. In other words, by means of the sales matrix, or a model based on it, we can estimate the future sales movements. This can be done either from the assumption that past relations will be kept approximately in the same proportions, or that they will change in the future. In each case all the changes can be described through the system based on the sales matrix.     

An algorithm for the calculation of transfer functions of two variable systems or networks

An algorithm is described to derive the transfer function of two variable networks or systems from the state equations. The algorithm is suitable for computer-aided procedures and gives recurrence relations from which the coefficients of the transfer function can be calculated. An alternative to this algorithm is to solve the state equations directly which requires a matrix inversion step at each frequency. By using the algorithm the coefficients of the transfer function are calculated only once and this offers substantial saving in computer time.     

The geometry of III-conditioning

Numerous problems in numerical analysis, including matrix inversion, eigen-value calculations, and polynomial zero finding, share the following property: the difficulty of solving a given problem is large when the distance from that problem to the nearest “ill-posed” one is small. For example, the closer a matrix is to the set of noninvertible matrices, the larger its condition number with respect to inversion. We show that the sets of ill-posed problems for matrix inversion, eigen-problems, and polynomial zero finding all have a common algebraic and geometric structure which lets us compute the probability distribution of the distance from a “random” problem to the set. From this probability distribution we derive, for example, the distribution of the condition number of a random matrix. We examine the relevance of this theory to the analysis and construction of numerical algorithms destined to be run in finite precision arithmetic.     

混凝土重力坝多参数弹性位移反演分析不唯一性理论探讨

反演分析是现场监测⁃反演分析⁃工程实践检验⁃正演分析及预测的闭环系统的重要环节,而参数反分析是工程实践中研究最多的反分析问题.针对混凝土重力坝多参数反演分析是否具有唯一性,基于均质地基上重力坝在水压力作用下的位移解析解建立目标函数,进而以目标函数和非空凸集构建一个凸规划问题,然后通过分析目标函数的Hesse矩阵是否是正定矩阵,验证目标函数是否是严格凸函数,从而辨识构建的凸规划问题是否具有唯一全局极小点.对坝体和岩基弹性参数的不同组合方案分析表明,当采用理论值与实测值的差值的l1范数作为目标函数时,目标函数的Hesse矩阵均不能保证为正定矩阵,即混凝土重力坝多参数弹性位移反演分析凸规划问题不具有唯一全局极小点,反演分析不唯一.     

A queueing model of delayed product differentiation

We consider a production system in which a supplier produces semi-finished items on a make-to-stock basis for a manufacturer that will customize the items on a make-to-order basis. The proportion of total processing time undertaken by the supplier determines how suitable the semi-finished items will be to meet customer demand. The manufacturer wishes to determine the optimal point of differentiation (the proportion of processing completed by the supplier) and its optimal semi-finished goods buffer size. We use matrix geometric methods to evaluate various performance measures for this system, and then, with enumeration techniques, obtain optimal solutions. We find that delayed product differentiation is attractive when the manufacturer can balance the costs of customer order fulfillment delay with the costs associated with unsuitable items.     

A single-period inventory placement problem for a supply chain with the expected profit objective

Consider the expected profit maximizing inventory placement problem in an N-stage, supply chain facing a stochastic demand for a single planning period for a specialty item with a very short selling season. Each stage is a stocking point holding some form of inventory (e.g., raw materials, subassemblies, product returns or finished products) that after a suitable transformation can satisfy customer demand. Stocking decisions are made before demand occurs. Because of delays, only a known fraction of demand at a stage will wait for shipments. Unsatisfied demand is lost. The revenue, salvage value, ordering, shipping, processing, and lost sales costs are proportional. There are fixed costs for utilizing stages for stock storage. After characterizing an optimal solution, we propose an algorithm for its computation. For the zero fixed cost case, the computations can be done on a spreadsheet given normal demands. For the nonnegative fixed cost case, we develop an effective branch and bound algorithm.     

The explicit derivation of the efficient portfolio frontier in the case of degeneracy and general singularity

In this paper the efficient portfolio frontier is derived explicitly for cases in which short sales are not allowed and more than one variable vanishes in a point of investment returns (degenerate case). When the covariance-variance matrix of the problem is singular (positive semi-definite) or diagonal some properties are also derived.     

Complex-valued Zhang neural network for online complex-valued time-varying matrix inversion

In this paper, a new complex-valued recurrent neural network (CVRNN) called complex-valued Zhang neural network (CVZNN) is proposed and simulated to solve the complex-valued time-varying matrix-inversion problems. Such a CVZNN model is designed based on a matrix-valued error function in the complex domain, and utilizes the complex-valued first-order time-derivative information of the complex-valued time-varying matrix for online inversion. Superior to the conventional complex-valued gradient-based neural network (CVGNN) and its related methods, the state matrix of the resultant CVZNN model can globally exponentially converge to the theoretical inverse of the complex-valued time-varying matrix in an error-free manner. Moreover, by exploiting the design parameter γ>1, superior convergence can be achieved for the CVZNN model to solve such complex-valued time-varying matrix inversion problems, as compared with the situation without design parameter γ involved (i.e., the situation with γ=1). Computer-simulation results substantiate the theoretical analysis and further demonstrate the efficacy of such a CVZNN model for online complex-valued time-varying matrix inversion.     

Wiener＇s lemma：localization and various approaches

Matrices and integral operators with off-diagonal decay appear in numerous areas of mathematics including numerical analysis and harmonic analysis, and they also play impor-tant roles in engineering science including signal processing and communication engineering. Wiener’s lemma states that the localization of matrices and integral operators are preserved un-der inversion. In this introductory note, we re-examine several approaches to Wiener’s lemma for matrices. We also review briefly some recent advances on localization preservation of operations including nonlinear inversion, matrix factorization and optimization.     

A class of block alternating splitting implicit iteration methods for double saddle point linear systems

In the framework of a special block alternating splitting implicit (BASI) iteration scheme for generalized saddle point problems, we establish some new iteration methods for solving double saddle point problems by means of a suitable partitioning strategy. Convergence analysis of the corresponding BASI iteration methods indicates that they are convergent unconditionally under certain weak requirements for the related matrix splittings, which are satisfied directly for our specific application to double saddle point problems. Numerical examples for liquid crystal director and time-harmonic eddy current models are presented to demonstrate the efficiency of the proposed BASI preconditioners to accelerate the GMRES method.     

推广成本信息不对称下推广商与酒店间的合作契约设计

为应对日趋激烈的酒店市场竞争、加强对销售渠道的控制,酒店急需与推广商合作以提高自有销售网站的需求水平。文章基于推广成本信息不对称,构建了一个占主导的酒店和一家推广商关于需求推广合作的博弈模型,分析了信息不对称情形下推广商的谎报动机,指出合作契约设计的必要性。并提出一套契约菜单,在保证推广商依据自身实际成本类型进行契约选择的前提下,实现了酒店利润的最大化。最后通过算例分析,验证了所设计的合作契约的有效性,并给出了参数灵敏性分析。     

Using the Sherman-Morrison-Woodbury inversion formula for a fast solution of tridiagonal block Toeplitz systems

A fast numerical algorithm for solving systems of linear equations with tridiagonal block Toeplitz matrices is presented. The algorithm is based on a preliminary factorization of the generating quadratic matrix polynomial associated with the Toeplitz matrix, followed by the Sherman-Morrison-Woodbury inversion formula and solution of two bidiagonal and one diagonal block Toeplitz systems. Tight estimates of the condition numbers are provided for the matrix system and the main matrix systems generated during the preliminary factorization. The emphasis is put on rigorous stability analysis to rounding errors of the Sherman-Morrison-Woodbury inversion. Numerical experiments are provided to illustrate the theory.     

与Bell多项式相关的一个新的矩阵反演及其应用

利用经典Lagrange反演公式, 本文给出了一个新的Bell矩阵反演, 由此建立了Bell多项式的一些新的性质, 其中包括一个Bell矩阵反演的封闭形式和经典Fa\`{a} di Bruno公式的一个逆形式.     

An inverse free parallel spectral divide and conquer algorithm for nonsymmetric eigenproblems

Summary. We discuss an inverse-free, highly parallel, spectral divide and conquer algorithm. It can compute either an invariant subspace of a nonsymmetric matrix , or a pair of left and right deflating subspaces of a regular matrix pencil . This algorithm is based on earlier ones of Bulgakov, Godunov and Malyshev, but improves on them in several ways. This algorithm only uses easily parallelizable linear algebra building blocks: matrix multiplication and QR decomposition, but not matrix inversion. Similar parallel algorithms for the nonsymmetric eigenproblem use the matrix sign function, which requires matrix inversion and is faster but can be less stable than the new algorithm. Received September 20, 1994 / Revised version received February 5, 1996     

Block triangular preconditioning for time-dependent Stokes control

We consider the numerical solution of time-dependent Stokes control problems, an important class of flow control problems within the field of PDE-constrained optimization. The problems we examine lead to large and sparse matrix systems which, with suitable rearrangement, can be written in block tridiagonal form, with the diagonal blocks given by saddle point systems. Using previous results for preconditioning PDE-constrained optimization and fluid dynamics problems, along with well-studied saddle point theory, we construct a block triangular preconditioner for the matrix systems. Numerical experiments verify the effectiveness of our solver. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Computational Solutions of Matrix Problems Over an Integral Domain

Recent methods for handling matrix problems over an integraldomain are investigated from a unifying point of view. Emphasizedare symbolic matrix inversion and numerically exact methodsfor solving Ax = b. New proofs are given for the theory of themultistep method. A proof for the existence and an algorithmfor the exact solution of Tx = b, where T is a finite Toeplitzmatrix, is given. This algorithm reduces the number of requiredsingle precision multiplications by a factor of order n overthe corresponding Gaussian elimination method. The use of residuearithmetic is enhanced by a new termination process. The matrixinversion problem with elements in the ring of polynomials isreduced to operations over a Galois field. It is shown thatinterpolation methods are equivalent to congruence methods withlinear modulus and that the Chinese remainder theorem over GF(x-p_k)is the Lagrange interpolation formula. With regard to the numerical problem of exact matrix inversion,the One- and Two-step Elimination methods are critically comparedwith the methods using modular or residue arithmetic. Formulasfor estimating maximum requirements for storage and timing ofthe salient parts of the algorithms are developed. The resultsof a series of recent tests, using existing codes, standardmatrices and matrices with random elements are reported andsummarized in tabular form. The paper concludes that the two-stepelimination method be used for the inversion problem of numericmatrices, and in particular when a black-box approach to thematrix inversion problem is attempted such as in commercialtime sharing systems. It is recommended that the inversion problemof matrices with elements over the polynomial ring be reducedto the numeric inversion problem with subsequent interpolation.An extensive Reference list is added.     

The Growth Matrix of the Sale

The model of the sale based on the growth matrix shows a double aspect of growth: one which is related to the direct growth rates and which expresses the intensity of sales, and another which is related to the indirect rates and shows a relative sales growth of one product in relation to another. The indirect sales growth provides us with knowledge about the relative relations of the sales growth. By simultaneous observation of the direct and indirect growth rates, we are in a position to find out the absolute and relative changes of sales during a period of time, and through the matrix of growth we present a complete structure of the sales growth.     

A summability method for the arithmetic Fourier transform

The Arithmetic Fourier Transform (AFT) is an algorithm for the computation of Fourier coefficients, which is suitable for parallel processing and in which there are no multiplications by complex exponentials. This is accomplished by the use of the Möbius function and Möbius inversion. However, the algorithm does require the evaluation of the function at an array of irregularly spaced points. In the case that the function has been sampled at regularly spaced points, interpolation is used at the intermediate points of the array. Generally theAFT is most effective when used to calculate the Fourier cosine coefficients of an even function.In this paper a summability method is used to derive a modification of theAFT algorithm. The proof of the modification is quite independent of theAFT itself and involves a summation by primes. One advantage of the new algorithm is that with a suitable sampling scheme low order Fourier coefficients may be calculated without interpolation.     

Fast algorithms for floating-point interval matrix multiplication

We discuss several methods for real interval matrix multiplication. First, earlier studies of fast algorithms for interval matrix multiplication are introduced: naive interval arithmetic, interval arithmetic by midpoint-radius form by Oishi-Rump and its fast variant by Ogita-Oishi. Next, three new and fast algorithms are developed. The proposed algorithms require one, two or three matrix products, respectively. The point is that our algorithms quickly predict which terms become dominant radii in interval computations. We propose a hybrid method to predict which algorithm is suitable for optimizing performance and width of the result. Numerical examples are presented to show the efficiency of the proposed algorithms.