Computational Methods for Structured and Singular Matrix Polynomials

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（DAEs）中，举几个例子。在相关的问题中，主要是正则矩阵多项式的特征结构的计算，以及在奇异多项式的情况下，最小指标和最小基的计算。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，从而构建了新的结构化线性化，浓缩形式和精确的结构保留算法。通过使用新技术，他们也在奇异多项式上取得了进展，表明线性化为最小指标和基的可靠计算提供了一条途径。这个建议挑出了几个重要的调查任务，涉及线性化、四项化和最小指数和基。目标是为这些计算开发新的算法，并增加理论理解，以帮助制定有效的算法。本方案所研究的问题普遍存在于工程和应用科学的许多重要问题中。数值方法的解决方案至关重要在结构力学,分子动力学、振动分析、模拟电路、各向异性材料的弹性变形,和光学波导设计,给几个例子。高速列车、光电设备、微机电系统和空客380等“超大型”喷气机等极端设计的趋势，对这些结构的谐振频率的计算提出了挑战。这些极端的设计通常会导致计算敏感的问题，而潜在问题的物理学导致了数值方法应该利用的结构，以获得物理上有意义的结果。该项目的目的是增加我们对保持这些结构的数学转换的理论理解，从而推进计算有效算法的发展。因此，这项工作将直接造福于各个学科的科学家和工程师。

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.