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(DAE)中，仅举几个例子。相关问题中的主要问题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，极小指数和极小基的计算。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，这些空间导致了新的结构化线性化、凝聚形式和精确的结构保持算法的构造。通过使用新技术，他们还在奇异多项式方面取得了进展，表明线性化提供了一条可靠计算最小指数和基的途径。这项建议列举了几项重要的调查任务，涉及线性化、平面化和最小指数和碱基。目标是为这些计算开发新的算法，并增加理论上的理解，以帮助形成有效的算法。在这个建议中研究的问题在工程和应用科学的广泛的重要问题中普遍存在。它们的数值求解方法在结构力学、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都是至关重要的，举几个例子。极端设计的趋势，如高速列车、光电子器件、微电子机械系统和空中客车380等超大型喷气式飞机，对这些结构的共振频率的计算提出了挑战。这些极端的设计往往导致计算敏感的问题，而潜在问题的物理导致结构，数值方法应该利用这些结构来获得物理上有意义的结果。这个项目的目的是增加我们对保留这些结构的数学变换的理论理解，从而促进计算有效算法的发展。因此，这项工作将直接惠及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.