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

提案#0098222格鲁吉亚州立大学Olshevsky，Vadim许多应用在科学和工程，以及在计算，应用和纯数学引起的问题涉及矩阵与Toeplitz，汉克尔，范德蒙德，柯西结构，沿着与许多其他模式的结构。标准数学软件工具[例如，MATLAB、Mathematica、MathCad、Maple、LAPACK]都是基于标准的[结构忽略]方法，因此它们的使用通常不适合获得满意的解决方案。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大量的CPU时间。其次，许多结构化矩阵是极端病态的，这意味着所有可用的标准方法可能无法在计算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algorithm.The本项目的目标是研究几个已知的和几个新的理论和计算问题的结构矩阵出现在几个应用领域，包括信号和图像处理，系统理论和控制理论。这项研究涉及到新的accuratefront和超快速算法的发展，在有理矩阵插值和逼近问题与规范约束（被动插值）中产生的几个新的类的结构矩阵。本研究也将集中于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.