Fast and Accurate Algorithms for Structured Matrix Computations: Applications and Software

建议#0098222佐治亚州立大学奥尔舍夫斯基，瓦迪姆在科学和工程以及计算、应用和纯数学方面的大量应用引起了涉及Toeplitz、Hankel、Vandermonde、CauchyStructures以及许多其他结构模式的矩阵的问题。标准的数学软件工具[例如，MatLab、MathCad、MathCad、Maple、LAPACK]都是基于标准的[结构忽略]方法，因此它们的使用往往不适合于获得满意的解。这里有两个主要原因。首先，忽略结构会人为地平方数据的大小，这需要不必要的存储和极其大量的CPU时间。其次，许多结构矩阵是极其病态的，这意味着所有可用的标准方法可能无法在计算的解中产生甚至一个正确的数字。克服这些困难的唯一方法是利用这种矩阵的特殊结构，并设计更有效的算法。本项目的目标是研究几个已知的和几个新的与结构化矩阵有关的理论和计算问题，这些问题出现在几个应用领域，包括信号和图像处理、系统论和控制论。本研究针对有理矩阵内插和范数约束(被动内插)逼近问题中出现的几类新的结构矩阵，提出了新的精确快速和超快算法。本文还将重点研究矩阵多项式的各种稳定性问题，以及纠错码理论中的各种问题。

Proposal #0098222Georgia State University Olshevsky, VadimNumerous applications in sciences and engineering, as well as in computational, applied and pure mathematics give rise to problems involving matrices with Toeplitz, Hankel, Vandermonde, Cauchystructures, along with many other patterns of structure. Standard mathematical software tools [e.g., MATLAB, Mathematica, MathCad, Maple, LAPACK], are based on standard [structure-ignoring]methods, and therefore their use is often not appropriate for obtaining a satisfactory solution. Here are two major reasons. First, ignoring the structure artificially squares the size of the data, which requires unnecessary storage and an extremely large amount of CPU time. Secondly, many structured matrices areextremely ill-conditioned, which means that all available standard methods may fail to produce even one correct digit in the computed solution. The only way to overcome these difficulties is to exploit the special structure of such matrices, and to design more efficient algorithms.The objective of this project is the study of several known and several new theoretical and computational problems related to structured matrices which arise in several applied areas, including signal and image processing, system theory, and control theory. This research involves the development of new accuratefast and superfast algorithms for several new classes of structured matrices arising in rational matrix interpolation and approximation problems with norm constraints (passive interpolation). This study will also focus on various stability problems for matrix polynomials, and on various problems in thetheory of error-correcting codes.