Computational Methods for Structured and Singular Matrix Polynomials

结构化和奇异矩阵多项式的计算方法

• 批准号: 1016224
• 负责人: D. Steven Mackey
• 金额: $ 30万
• 依托单位: Western Michigan University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-07-01 至 2014-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1016224&HistoricalAwards=false
• 关键词: Computational; Methods; Structured; Singular; Matrix

Matrix polynomials frequently arise in the engineering and applied sciences,especially in structural dynamics, vibrational analysis, control systems, and differential-algebraic equations (DAEs), to give a few examples. Principal among the associated problems are the computation of the eigenstructure of regular matrix polynomials, and in the case of singular polynomials, the computation of minimal indices and minimal bases. In recent work, the investigators and their colleagues identified rich spaces of linearizations which led to the construction of new structured linearizations,condensed forms, and accurate structure-preserving algorithms. By using new techniques, they have also made progress on singular polynomials, showing that linearizations provide a pathway to the reliable computation of minimal indices and bases. This proposal singles out several important tasks for investigation concerning linearizations, quadratifications and minimal indices and bases. The goal is to develop new algorithms for these computations, and increase theoretical understanding so as to aid in the formulation of effective algorithms.The problems studied in this proposal are ubiquitous in a wide range of important problems in engineering and applied sciences. Numerical methods for their solution are critical in structural mechanics, molecular dynamics, vibrational analysis, the simulation of electrical circuits, elastic deformation of anisotropic materials, and optical waveguide design, to give a few examples. The trend towards extreme designs, such as high speed trains, optoelectronic devices, micro-electromechanical systems, and ``superjumbo'' jets such as the Airbus 380, presents a challenge for the computation of the resonant frequencies of these structures. These extreme designs often lead to computationally sensitive problems, while the physics of the underlying problem leads to structure that numerical methods should exploit in order to obtain physically meaningful results. The aim of this project is to increase our theoretical understanding of mathematical transformations that preserve these structures and thereby advance the development of computationally effective algorithms. Consequently, this work will have direct benefit to scientists and engineers across a wide range of disciplines.

矩阵多项式经常出现在工程和应用科学中，特别是在结构动力学、振动分析、控制系统和微分代数方程（DAE）中，仅举几个例子。主要的相关问题是计算的特征结构的正规矩阵多项式，并在奇异多项式的情况下，计算的最小指数和最小基地。在最近的工作中，研究人员和他们的同事确定了丰富的线性化空间，这导致了新的结构化线性化，压缩形式和精确的结构保持算法的构建。通过使用新技术，他们在奇异多项式上也取得了进展，表明线性化提供了一条可靠计算最小指数和基的途径。这个建议挑出几个重要的任务，调查有关线性化，quadratification和最小指数和基地。我们的目标是为这些计算开发新的算法，并增加理论上的理解，以帮助制定有效的algorithm.The问题研究在这个建议是无处不在的，在工程和应用科学的广泛的重要问题。在结构力学、分子动力学、振动分析、电路模拟、各向异性材料的弹性变形和光波导设计中，数值方法是至关重要的。极端设计的趋势，如高速列车，光电设备，微机电系统，和“超级巨型”喷气式飞机，如空中客车380，提出了一个挑战，这些结构的谐振频率的计算。这些极端的设计通常会导致计算敏感的问题，而潜在问题的物理特性会导致数值方法应该利用的结构，以获得物理上有意义的结果。这个项目的目的是增加我们的数学变换，保持这些结构的理论理解，从而推进计算有效的算法的发展。因此，这项工作将直接造福于各个学科的科学家和工程师。