Remarks concerning the optimization of matrix codes

This paper describes the results of a general theory of matrix codes correcting a set of given types of multiple errors. A detailed study has been made of certain matrix classes of these systematic binary error correcting codes that will correct typical errors of some digital channels. These codes published by Elias,^(2,3) Hobb's,⁽⁵⁾ and Voukalis⁽¹¹⁾ account for this theory and other new families of binary systematic matrix codes of arbitrary size, correcting random, burst and clusters of errors are given here. Also presented here are the basic ideas of each of these codes. We can easily find practical decoding algorithms for each of these codes. The characteristic calculation of the parity check equations that the information matrix codebook has to satisfy are also shown. Further on we deal with the optimum construction of these codes showing their use in certain applications. We answer questions such as: “What is the optimum size of the code?” “What is the best structure of the code?” “What is the probability of error correction and the mean error correction performance?” Consequently, in this paper we also describe the results of an extensive search for optimum matrix codes designed to correct a given set of multiple errors as well as their implementation.     

Maple code for the calculation of the matrix elements of the Boltzmann collision operators for mixtures

A Maple code is provided which is used to compute the matrix elements of the collision operators in the Boltzmann equation for arbitrary differential elastic collision cross section. The present paper describes an efficient method for the calculation of the matrix elements of the collision operators in the Sonine basis set. The method employs the generating functions for these polynomials. The transport properties of gaseous mixtures of atoms and/or ions are generally determined from solutions of the Boltzmann equation. The solution of the Boltzmann equation for the velocity distribution functions requires a representation of the integral collision operators defined by the differential cross sections describing collisions between pairs of particles. Many applications have considered either the simple hard sphere cross section or the cross section corresponding to the inverse fourth power of the inter-particle distance (“Maxwell molecules”). There are a few applications where realistic quantum mechanical cross sections have been used. The basis set of Sonine (or Laguerre) polynomials is the basis set of choice used to represent the distribution functions. The Maple code provided is used to express the matrix elements of the collision operators in terms of a finite sum of the omega integrals of transport theory and defined by the differential cross section. Thus the matrix representations of the collision operators are applicable to arbitrary interaction potentials.

Program summary

Program title: MCBCCatalogue identifier: AEGJ_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEGJ_v1_0.htmlProgram obtainable from: CPC Program Library, Queen's University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 2422No. of bytes in distributed program, including test data, etc.: 48 653Distribution format: tar.gzProgramming language: MapleComputer: Personal computer or workstationOperating system: Windows or LinuxRAM: bytesClassification: 12Nature of problem: The representation of the collision operators in the Boltzmann equation in the basis set of Sonine (Laguerre) polynomials.Solution method: The matrix element of the generating function for the Sonine polynomials are evaluated analytically. The individual matrix elements are the coefficients in the double power series expansion in two parameters. The Maple code is used to effect this expansion and extract the coefficients of the omega integrals. The omega integrals are defined by the differential cross section.Running time: A few seconds for the examples provided.     

On the comparisons of the multisplitting unsymmetric aor methods for M-matrices

This paper reveals the inner links between two known frameworks of multisplitting relaxation methods as completely as possible. By meticulously investigating the specific structures of these two frameworks, the asymptotic convergence rates as well as the monotone convergence rates of them are compared theoretically. This then definitely answers the question that which converges faster between these two frameworks of parallel matrix multisplitting relaxation methods from the standpoint of pure mathematics. At last, an example further confirms the correctness of the theoretical results.     

Lower bounds for the multiplicative complexity of matrix multiplication

We prove a lower bound of km + mn + k—m + n— 3 for the multiplicative complexity of the multiplication of -matrices with -matrices using the substitution method. Received: July 8, 1997.     

LDPC比特翻转译码算法的分析与改进

利用统计译码思想由LDPC（Low Density Parity Check）码校验矩阵通过矢量的线性组合构造出一个新的低密度校验矢量集合，并结合LDPC码并行比特翻转译码算法的环检测等特点的分析，提出了一种新的硬判决译码方案。仿真结果表明：改进算法在译码性能上接近BP算法，又保持了并行比特翻转算法迭代次数少的优点。     

Subspace learning for unsupervised feature selection via matrix factorization

Dimensionality reduction is an important and challenging task in machine learning and data mining. Feature selection and feature extraction are two commonly used techniques for decreasing dimensionality of the data and increasing efficiency of learning algorithms. Specifically, feature selection realized in the absence of class labels, namely unsupervised feature selection, is challenging and interesting. In this paper, we propose a new unsupervised feature selection criterion developed from the viewpoint of subspace learning, which is treated as a matrix factorization problem. The advantages of this work are four-fold. First, dwelling on the technique of matrix factorization, a unified framework is established for feature selection, feature extraction and clustering. Second, an iterative update algorithm is provided via matrix factorization, which is an efficient technique to deal with high-dimensional data. Third, an effective method for feature selection with numeric data is put forward, instead of drawing support from the discretization process. Fourth, this new criterion provides a sound foundation for embedding kernel tricks into feature selection. With this regard, an algorithm based on kernel methods is also proposed. The algorithms are compared with four state-of-the-art feature selection methods using six publicly available datasets. Experimental results demonstrate that in terms of clustering results, the proposed two algorithms come with better performance than the others for almost all datasets we experimented with here.     

Monte Carlo methods for matrix computations on the grid

Many scientific and engineering applications involve inverting large matrices or solving systems of linear algebraic equations. Solving these problems with proven algorithms for direct methods can take very long to compute, as they depend on the size of the matrix. The computational complexity of the stochastic Monte Carlo methods depends only on the number of chains and the length of those chains. The computing power needed by inherently parallel Monte Carlo methods can be satisfied very efficiently by distributed computing technologies such as Grid computing. In this paper we show how a load balanced Monte Carlo method for computing the inverse of a dense matrix can be constructed, show how the method can be implemented on the Grid, and demonstrate how efficiently the method scales on multiple processors.     

Riccati discrete time transfer matrix method for elastic beam undergoing large overall motion

An efficient method for dynamics simulation for elastic beam with large overall spatial motion and nonlinear deformation, namely, the Riccati discrete time transfer matrix method (Riccati-DT-TMM), is proposed in this investigation. With finite segments, continuous deformation field of a beam can be decomposed into many rigid bodies connected by rotational springs. Discrete time transfer matrices of rigid bodies and rotational springs are used to analyze the dynamic characteristic of the beam, and the Riccati transform is used to improve the numerical stability of discrete time transfer matrix method of multibody system dynamics. A predictor-corrector method is used to improve the numerical accuracy of the Riccati-DT-TMM. Using the Riccati-DT-TMM in dynamics analysis, the global dynamics equations of the system are not needed and the computation time required increases linearly with the system’s number of degrees of freedom. Three numerical examples are given to validate the method for the dynamic simulation of a geometric nonlinear beam undergoing large overall motion.     

Characterizations for the regular prefix codes and related families

Intrinsic characterizations by means of analogues of regular expressions are given for six families of regular languages related to the prefix codes, namely their reversals and their closure under union, the right and left ideals, and their complements. First, a characterization for the regular prefix codes is obtained, which is then used to characterize the other families. Characterizations by finite automata are also presented.This research has been sponsored in part by FINEP, CNPq and Canadian NRC.     

A unified framework for the construction of various matrix multisplitting iterative methods for large sparse system of linear equations

A unified framework for the construction of various synchronous and asynchronous parallel matrix multisplitting iterative methods, suitable to the SIMD and MIMD multiprocessor systems, respectively, is presented, and its convergence theory is established under rather weak conditions. These afford general method models and systematical convergence criterions for studying the parallel iterations in the sense of matrix multisplitting. In addition, how the known parallel matrix multisplitting iterative methods can be classified into this new framework, and what novel ones can be generated by it are shown in detail.     

An iterative algorithm for solving a pair of matrix equations over generalized centro-symmetric matrices

A matrix is said to be a symmetric orthogonal matrix if . A matrix is said to be generalized centro-symmetric (generalized central anti-symmetric) with respect to P, if A=PAP (A=−PAP). The generalized centro-symmetric matrices have wide applications in information theory, linear estimate theory and numerical analysis. In this paper, we propose a new iterative algorithm to compute a generalized centro-symmetric solution of the linear matrix equations . We show, when the matrix equations are consistent over generalized centro-symmetric matrix Y, for any initial generalized centro-symmetric matrix Y₁, the sequence {Y_k} generated by the introduced algorithm converges to a generalized centro-symmetric solution of matrix equations . The least Frobenius norm generalized centro-symmetric solution can be derived when a special initial generalized centro-symmetric matrix is chosen. Furthermore, the optimal approximation generalized centro-symmetric solution to a given generalized centro-symmetric matrix can be derived. Several numerical examples are given to show the efficiency of the presented method.     

Improvement in the picture quality of moving pictures for matrix displays

Abstract— When a picture moves linearly at a specific velocity, its spectrum is distributed in the gaps between the high‐order harmonics in the 3‐D spatial‐frequency region, and therefore the pass band of the picture spectrum is apparently extended. The extension of the pass band does not contribute to picture quality in the RGB‐trio arrangement, but in some mosaic arrangements it may improve picture quality by eliminating discoloring of details due to human‐vision characteristics. The relation between picture motion and quality, which can be represented by the 3D Nyquist limits, can be verified by experiment or by the simple TFZP method. Subjective evaluation of the RGGB‐mosaic shows that, when the picture moves linearly, discoloring is almost completely eliminated at a velocity that can be estimated from the structure of the 3D Nyquist limits incorporating human‐vision characteristics. Good picture quality can be obtained at a velocity of 1 dot/field. The TFZP method makes it possible to verify the region where discoloring is eliminated by directly observing a cross‐section of the 3D Nyquist limits that incorporates human‐vision characteristics.     

基于改进自由权矩阵方法的神经网络时滞相关稳定性分析

针对具有不确定参数的时变时滞神经网络系统,利用改进的自由权矩阵方法,研究其时滞相关稳定性问题。通过考虑时变时滞及其上界和它们的差三者之间关系,同时保留Lyapunov—Krasovskii泛函导数中的有用项,得到具有更低保守性的基于线性矩阵不等式的神经网络系统时滞相关渐近稳定的充分条件。最后,数值例子表明该方法的有效性。     

A multiplicative multisplitting method for solving the linear complementarity problem

The convergence of the multiplicative multisplitting-type method for solving the linear complementarity problem with an H-matrix is discussed using classical and new results from the theory of splitting. This directly results in a sufficient condition for guaranteeing the convergence of the multiplicative multisplitting method. Moreover, the multiplicative multisplitting method is applied to the H-compatible splitting and the multiplicative Schwarz method, separately. Finally, we establish the monotone convergence of the multiplicative multisplitting method under appropriate conditions.     

A new extended matrix logarithm formulation for the simulation of viscoelastic fluids by spectral elements

This study presents a new algorithm developed in order to remove instabilities observed in the simulation of unsteady viscoelastic fluid flows in the framework of the spectral element method. In this study, we consider a particular model of the finite extensible nonlinear elastic family, FENE-P, but the method could be applied to other differential constitutive equations. Two distinct constraints for the FENE-P equation are imposed: (i) the square of the corresponding finite extensibility parameter of the polymer must be an upper limit for the trace of the conformation tensor and (ii) the eigenvalues of the conformation tensor should remain positive at all steps of the simulation. Negative eigenvalues cause the unbounded growth of instabilities in the flow. The proposed transformation is an extension of the matrix logarithm formulation originally presented by Fattal and Kupferman [1] and [2]. To evaluate the capability of this new algorithm with the classical conformation tensor, comprehensive studies have been done based on the linear stability analysis to show the influence of this method on the resulting eigenvalue spectra and explain its success to tackle high Weissenberg numbers. With this new method one can tackle high Weissenberg number flow at values of practical interest. A neat improvement of the computational algorithm with stable convergence has been demonstrated in this study.     

On the convergence of the Bl-LSQR algorithm for solving matrix equations

Karimi and Toutounian [The block least squares method for solving nonsymmetric linear system with multiple right-hand sides, Appl. Math. Comput. 177 (2006), pp. 852–862], proposed a block version of the LSQR algorithm, say Bl-LSQR, for solving linear system of equations with multiple right-hand sides. In this paper, the convergence of the Bl-LSQR algorithm is studied. We deal with some computational aspects of the Bl-LSQR algorithm for solving matrix equations. Some numerical experiments on test matrices are presented.     

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.     

Development of a mixed control volume – Finite element method for the advection–diffusion equation with spectral convergence

In this paper we attack the problem of devising a finite volume method for computational fluid dynamics and related phenomena which can deal with complex geometries while attaining high-orders of accuracy and spectral convergence at a reasonable computational cost. As a first step towards this end, we propose a control volume finite element method for the solution of the advection–diffusion equation. The numerical method and its implementation are carefully tested in the paper where h- and p-convergence are checked by comparing numerical results against analytical solutions in several relevant test-cases. The numerical efficiency of a selected set of operations implemented is estimated by operation counts, ill-conditioning of coefficient matrices is avoided by using an appropriate distribution of interpolation points and control-volume edges.