CAREER: Studies in Numerical Linear Algebra

该项目将为工程计算中出现的线性代数问题开发快速准确的数值算法。一类这样的问题涉及具有位移结构的矩阵。这是一个一般的代数概念，包括著名的Toeplitz， Hankel， Vandermonde和Cauchy矩阵。在实践中，需要快速和准确地求解涉及这种矩阵的精确确定和过度确定的线性系统。此外，在一些应用中，如地球物理问题的逆散射，也需要这种矩阵的快速分解。另一类相关的问题出现在自适应滤波应用中，其中需要快速准确地解决由低秩变化相关的一系列结构化线性问题。另一方面，对称铅笔的特征值和特征向量的计算是另一个重要问题。它经常出现在工程结构的有限元分析和相关的模型简化问题中。新的模型和快速算法分析线性代数问题与大有界不确定性也将发展。

This project will develop fast and accurate numerical algorithms for linear algebra problems arising in engineering computations. One class of such problems concerns matrices with displacement structure. This is a general algebraic concept and includes the well-known classes of Toeplitz, Hankel, Vandermonde and Cauchy matrices. In practice there is the need for solving rapidly and accurately, both exactly-determined and over-determined linear systems involving such matrices. Furthermore, in some applications like inverse scattering for geophysical problems, there is also the need for the rapid factorization of such matrices. Another related class of problems arises in adaptive filtering applications, where there is a need to solve rapidly and accurately a sequence of structured linear problems that are related by low-rank changes. In a different vein, the computation of the eigenvalues and eigenvectors of symmetric pencils is another important problem. It frequently arises in finite-element analysis of engineering structures and associated model-reduction problems. New models and fast algorithms for the analysis of linear algebra problems with large bounded uncertainties will also be developed.