Fast and Numerically Accurate Algorithms for Matrices with Displacement Structure

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，Toeplitz加汉克尔，范德蒙结构，可控性，可观性，柯西，Loewner，皮克，舒尔-科恩，Routh-Hurwitz矩阵，Bezoutians，沿着与许多其他模式的结构。 标准数学软件工具（例如，MATLAB、MathCad、Maple V、NAG library、LAPACK）基于标准（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，例如Hankel，Pick，Hilbert，Vandermonde，都是极端病态的，这意味着所有可用的标准方法通常无法在计算的解中产生甚至一个正确的数字。 该项目的主题是研究与柯西和类柯西矩阵相关的几个理论和计算问题，这些问题出现在许多应用领域，包括信号处理，系统理论H ∞控制。 利用位移结构的概念，可以将柯西矩阵的算法用于设计其他重要类型结构矩阵的有效实用算法，如Toeplitz矩阵、Hankel矩阵、Toeplitz + Hankel矩阵、Vandermonde矩阵、Chebyshev-Vandermonde矩阵等。实际上，所有这些矩阵都可以通过应用一些快速傅立叶变换，正弦变换的余弦 它计划扩展的结果，以开发快速和数值精确的算法，新的结构矩阵类。 ***

Numerous applications in sciences and engineering, as well as in computational, applied and pure mathematics give rise to problems involving matrices with Toeplitz, Hankel, Toeplitz-plus-Hankel, Vandermonde structures, controllability, observability, Cauchy, Loewner, Pick, Schur-Cohn, Routh-Hurwitz matrices, Bezoutians, along with many other patterns of structure. standard mathematical software tools (e.g., MATLAB, MathCad, Maple V, NAG library, 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, e.g. Hankel, Pick, Hilbert, Vandermonde, are extremely ill- conditioned, which means that all available standard methods often fail to produce even one correct digit in the computed solution. The theme of this project is the study of several theoretical and computational problems related to Cauchy and Cauchy-like matrices, which arise in many applied areas, including signal processing, system Theory H-infinity control. Use of the concept of displacement structure suggests that algorithms for Cauchy-like matrices can be used to design efficient practical algorithms for many other important kinds of structured matrices, such as Toeplitz, Hankel, Toeplitz-plus- Hankel, Vandermonde, Chebyshev-Vandermonde matrices, etc. In fact, all these matrices can be transformed into Cauchy-like matrices just by applying a few fast Fourier, cosine of sine transforms. It is planned to extend the results to develop fast and numerically accurate algorithms for new classes of structured matrices. ***