Parallel solvers for discrete-time algebric Riccati equations

We investigate the numerical solution of discrete-time algebraic Riccati equations on a parallel distributed architecture. Our solvers obtain an initial solution of the Riccati equation via the disc function method, and then refine this solution using Newton's method. The Smith iteration is employed to solve the Stein equation that arises at each step of Newton's method. The numerical experiments on an Intel Pentium-II cluster, connected via a Myrinet switch, report the performance and scalability of the new algorithms. Copyright © 2001 John Wiley & Sons, Ltd.     

On an energy minimizing basis for algebraic multigrid methods

This paper is devoted to the study of an energy minimizing basis first introduced in Wan, Chan and Smith (2000) for algebraic multigrid methods. The basis will be first obtained in an explicit and compact form in terms of certain local and global operators. The basis functions are then proved to be locally harmonic functions on each coarse grid element. Using these new results, it is illustrated that this basis can be numerically obtained in an optimal fashion. In addition to the intended application for algebraic multigrid method, the energy minimizing basis may also be applied for numerical homogenization.     

Common subexpression processing in multiple-query processing

The efficiency of common subexpression identification is critical to the performance of multiple-query processing. In this paper, we develop a multigraph for representing and facilitating the processing of multiple queries. In addition to the traditional multiple-query processing approaches in exploiting common subexpressions for identical and subsumption cases, the proposed multigraph processing also covers the overlap case. A performance study shows the viability of this technique when compared to an earlier multigraph approach     

Augmenting a (k - 1)-Vertex-Connected Multigraph ℓ-Edge-Connected and k-Vertex-Connected Multigraph

For two integers $k,\ell > 0$ and an undirected multigraph $G=(V,E)$, we consider the problem of augmenting $G$ by the smallest number of new edges to obtain an $\ell$-edge-connected and $k$-vertex-connected multigraph. In this paper we show that a $(k-1)$-vertex-connected multigraph $G$ can be made $\ell$-edge-connected and $k$-vertex-connected by adding at most $\bound$ surplus edges over the optimum in $O(\tm)$ time, where $n=|V|$.     

一种基于结构分解和因子分析的贝叶斯网络隐变量发现算法

通过研究粗糙集与图论的关系,提出了以集合为权的加权多重完全多部图的概念,定义了加权多重完全多部图的邻接矩阵,得到了加权完全多部图与决策表的映射关系;给出了粗糙集决策表信息系统的图论形式和决策表信息系统属性约简的图论方法,并根据图论理论对算法进行了优化;得到了在决策表信息系统中,属性的集合不可以约简的充分必要条件;并进一步提出了基于属性置信度的计算方法和多决策属性的处理方法。编程实验结果证明该方法能有效地降低时间和空间复杂度。     

Stability of quaternionic linear systems

The main goal of this paper is to characterize stability and bounded-input-bounded-output (BIBO)-stability of quaternionic dynamical systems. After defining the quaternion skew-field, algebraic properties of quaternionic polynomials such as divisibility and coprimeness are investigated. Having established these results, the Smith and the Smith-McMillan forms of quaternionic matrices are introduced and studied. Finally, all the tools that were developed are used to analyze stability of quaternionic linear systems in a behavioral framework.     

Error Analysis and Comparison of Two Algorithms Measuring Compensated Income

We analyze and compare the errors of two numerical approaches for measuring compensated income. We prove that Vartia’s algorithm and Breslaw and Smith’s algorithm both converge quadratically; when the price change within each partition step is small, the error of Vartia’s algorithm is approximately half that of Breslaw and Smith’s algorithm. A theorem and the appropriate simulations with three different demand systems correct the existing error analysis of the two algorithms, and demonstrate that Vartia’s algorithm is more accurate than Breslaw and Smith’s.     

An algebraic variational multiscale–multigrid method for large eddy simulation of turbulent flow

An algebraic variational multiscale–multigrid method is proposed for large eddy simulation of turbulent flow. Level-transfer operators from plain aggregation algebraic multigrid methods are employed for scale separation. In contrast to earlier approaches based on geometric multigrid methods, this purely algebraic strategy for scale separation obviates any coarse discretization besides the basic one. Operators based on plain aggregation algebraic multigrid provide a projective scale separation, enabling an efficient implementation of the proposed method. The application of the algebraic variational multiscale–multigrid method to turbulent flow in a channel produces results notably closer to reference (direct numerical simulation) results than other state-of-the-art methods both for mean streamwise and root-mean-square velocities. For predicting highly sensitive components of the Reynolds-stress tensor in the context of turbulent recirculating flow in a lid-driven cavity, the algebraic variational multiscale–multigrid method also shows a remarkably good performance in predicting reference results from experiment and direct numerical simulation compared to other methods.     

A modified Runge-Kutta method with phase-lag of order infinity for the numerical solution of the Schrödinger equation and related problems

A modified Runge-Kutta method with phase-lag of order infinity for the numerical solution of the Schr?dinger equation and related problems is developed in this paper. This new modified method is based on the classical Runge-Kutta method of algebraic order four. The numerical results indicate that this new method is more efficient for the numerical solution of the Schr?dinger equation and related problems than the well known classical Runge-Kutta method of algebraic order four.     

Detecting common subexpressions for multiple query optimization over loosely-coupled heterogeneous data sources

The research presented in this paper supports the identification of common subexpressions as candidates for potential materialized views that form the basis of multiple query optimization in a loosely-coupled distributed system where query expressions access heterogeneous data sources, including relations and data-centric XML. This paper introduces a unifying mixed multigraph formalism to represent SQL, XQuery, and LINQ queries in a common query graph model and a heuristics-based algorithm to detect common subexpressions. The identified common subexpressions represent an opportunity for defining a materialized view to avoid repeating computation. The common subexpressions may access only relations, only XML, or a combination of relations and XML. The mixed multigraph model and the heuristic rules presented in this paper have distinguished advantages over the existing approaches that consider only relational or XML data sources individually. The mixed multigraph model can present SQL, XQuery, and LINQ queries in a single graph model and the heuristic rules are designed to consider the identical and subsumed conditions at the same time. A prototype implementation of the algorithm illustrates the applicability of the approach using various examples from the research literature as well as scenarios over a Criminal Justice enterprise that include common subexpressions across relational and XML data sources.     

连续属性决策表离散化的图论方法

通过研究粗糙集与图论的关系,提出了以集合为权的加权多重完全多部图的概念,得到了加权完全多部图与连续属性决策表的映射关系,给出了断点重要性和断点效率的一种新的量化定义并得到了相关性质;提出了连续属性决策表信息系统的图论形式和连续属性决策表离散化的图论方法。编程实验结果证明,应用此方法可以确保在离散化后决策表相容的前提下得到无剩余属性值的较小的断点集合。     

Matrix corrections minimal with respect to the Euclidean norm for linear programming problems

We consider special algebraic constructions, namely minimal matrix solutions and corrections of systems of linear algebraic equations and pairs of conjugate systems of linear algebraic equations. We build the corresponding mathematical apparatus that also lets us solve inverse linear programming problems (construct model linear programming problems with given properties), study and solve approximate and singular linear programming problems. We give statements of the theorems and numerical examples.     

Fortran to Lisp Translation using f2cl

The need for both algebraic and numerical capabilities within mathematical computation systems has highlighted the need to translate numerical software written in Fortran77 to Common Lisp, a language favoured by the algebraic computation community. This paper reports the current state of f2cl, a translator written to achieve that end. We describe the translation process giving details of Lisp equivalents of Fortran expressions, as well as discussing features of Fortran that have no straightforward equivalent in Lisp.     

Methods of solving a polynomial equation for anH∞ optimal control problem for a single-inputsingle-output discrete-time system

A polynomial equation for the H_∞ optimal control problem is reduced to a nonlinear algebraic equation. Two methods are proposed for solving the algebraic equation. One method uses the singularity of a linear algebraic equation as the optimality index. The other gives an approximate solution by solving an eigenvalue problem. A numerical example is presented     

2-D常系数线性离散系统可分性判定

针对二维常系数线性离散系统一般状态空间模型(2-DGM)的可分性判定问题进行探讨,给出了系统可分的若干判定法则,这些法则有助于进一步完善2-D系统现有的理论体系.     

STWS approach for Hammerstein system of nonlinear Volterra integral equations of the second kind

In this paper, single-term Walsh series (STWS) method is applied to obtain the numerical solutions of Hammerstein systems of nonlinear Volterra integral equations of second kind (HSNVIES). Using the properties of the STWS method, HSNVIES can be easily converted into solvable recursive system of algebraic equations. Solutions obtained from the recursive system of algebraic equations are the solutions of the HSNVIES. Illustrative examples are provided with numerical solutions and the efficiency of this STWS method is also compared with the existing methods.     

基于光滑聚集代数多重网格的有限元 并行计算实现方法

基于光滑聚集代数多重网格法实现一种用于结构有限元并行计算的预条件共轭梯度求解方法。对计算区域进行均匀划分,将这些子区域分配给各个进程同时进行单元刚度矩阵的计算,并组合形成分布式存储的整体平衡方程。采用光滑聚集代数多重网格预条件共轭梯度法对整体平衡方程进行并行求解,在天河二号超级计算机上进行数值试验,分析代数多重网格的主要参数对算法性能的影响,测试程序的并行计算性能。试验结果表明该方法具有较好的并行性能和可扩展性,适合于大规模实际应用。     

具有巴黎期权特性的可转债定价问题研究

采用有限差分方法对基于Black—Scholes方程的可转债定价模型进行数值求解,用Euler-Lagrange分裂格式离散包含具有巴黎期权特性的赎回条款的修正Black—Scholes方程,并以工行转债和中行转债的历史数据为例,比较不同的数学模型中定价结果与实际价格的差异,分析标的股票处在不同价位水平时不同定价模型对可转债问题的适用性．     

Scaling of general quadratic matrix equations

A scaling framework for general quadratic algebraic matrix equations is presented. All algebraic quadratic equations can be considered as special cases of a single generalized algebraic quadratic matrix equation (GQME). Hence, the paper is focused on the analysis and solution of the scaling problem of that GQME. The presented scaling method is based on the assignment of predetermined values of the coefficients and the unknown matrices of the GQME. The proposed framework is independent of any numerical method and therefore its use is general. Implementations are presented for the special case of matrix algebraic Riccati equations (AREs). Some new results of matrix algebraic identities considering Kronecker and Hadamard products are also reported.     

Derivation and implementation of an algorithm for singular integrals

A numerical construction of extended Gaussian quadrature rules for weight functions with algebraic and logarithmic singularities is presented. A computer program is described and numerical examples are given.