CAREER: Algorithms for Eigenvalue and Singular Value Problems

该CAREER项目的研究部分为许多重要问题开发了准确和有效的数值方法，包括高相对精度（微小百分比误差）特征值和奇异值计算，更新奇异值分解（SVD）以及线性系统和控制计算中的距离问题。尽管目前解决这些问题的方法很多，但许多重要问题仍未得到解答。例如，最近的大量论文描述了对少数孤立的特殊类型矩阵计算所有特征值和奇异值的数值方法，但缺乏一个通用的解释和通用的数值方法。作为另一个例子，快速更新SVD的能力在信号处理应用中是至关重要的，因为问题通常是实时的，但目前的方法都不够有效。目的是为高相对精度的特征值和奇异值计算提供一个通用的解释和通用的数值方法。事实证明，通用的解释和通用的数值方法适用于许多新的有趣的矩阵类，它们产生于从组合学到ODE和PDE的数值解等不同的领域。该项目还旨在提供快速更新SVD的方法，从而消除基于SVD的方法在各种信号处理应用中的主要瓶颈。该项目的教育部分是开发软件工具，使参加抽象数学课程的学生能够清楚地想象和玩在万维网上有时抽象的对象和结果，并以生动的图形显示。该计划还包括具有科学和工程应用的数值线性代数的新研究生课程。

The research part of this CAREER project develops accurate and efficient numerical methods for a number of important problems including high relative accuracy (tiny percentage error) eigenvalue and singular value computations, updating the singular value decomposition (SVD), and distance problems in linear systems and control computations. Although current methods for solving them are abundant, many important questions remain unanswered. For example, a large number of recent papers describe numerical methods to compute all eigenvalues and singular values to high relative accuracy for a few isolated special classes of matrices, but a common explanation and a common numerical method are missing. As another example, the ability to rapidly update the SVD is critical in signal processing applications since problems are often real- time, but none of the current methods is efficient enough for this purpose. A goal is to provide a common explanation and a common numerical method for high relative accuracy eigenvalue and singular value computations. It turns out that the common explanation and common numerical method are applicable to many new interesting classes of matrices, arising from areas as diverse as combinatorics and numerical solution of ODE's and PDE's. The project also aims at providing rapid methods for updating the SVD, hence removing a major bottleneck in SVD-based methods for various signal processing applications. The education part of the project is to develop software tools so that students taking abstract mathematics courses can clearly envision and play with the sometimes abstract objects and results on the World-Wide Web with vivid graphical display. The plan also includes new graduate courses in numerical linear algebra with scientific and engineering applications.