Matrix Functions, Rational Approximation, and Quadrature with Applications

本研究计画系关于发展与分析矩阵函数与由矩阵函数所定义之积分之逼近新方法。该研究融合了线性代数和逼近理论。新的合理的Lanczos方法埃尔米特矩阵的短递归公式正在开发中。短递归关系的存在性具有重要的理论和实际意义。它加快了计算速度，因为每个新的有理Lanczos向量只需要与最近计算的几个Lanczos向量正交。 所需的显式正交化的数量是有界的理性Lanczos步骤的数量无关。递归关系定义了一个给定的矩阵到一个有理Krylov子空间的正交投影。投影由五对角矩阵表示，该矩阵也具有块结构。该矩阵类似于与（标准）高斯求积规则相关联的对称三对角矩阵，并且是用于评估有理高斯、高斯-Radau规则和高斯-Lobatto求积规则的新算法的基础。这项研究是一个扩展的非常重要的工作基因Golub，和他的合作者，对矩阵，矩，正交多项式，和正交。可能存在与结构化矩阵的连接，例如在单位圆上的正交多项式和正交有理函数的背景下出现的CMV矩阵，以及与半可分矩阵和拟可分矩阵的连接。当给定的矩阵是非Hermitian时，可以应用有理Arnoldi和有理非Hermitian Lanczos过程。有理Krylov子空间上的投影也有一个结构，利用它是线性代数中的一个重要课题。在许多应用中需要估计矩阵泛函，包括社交网络的研究和不适定逆问题的Tikhonov正则化。 主要研究员正在研究更快的数值方法的发展，用于评估，限制或估计涉及大型对称或非对称矩阵的复杂非线性表达式。速度的提高是通过利用迄今为止一直被忽视的结构来实现的。在解决科学家和工程师感兴趣的大规模问题时，快速方法的发展是重要的。这些方法适用于社交网络的调查，其属性最近得到了相当大的关注，不仅是科学家，而且由纽约时报。研究社交网络的一个主要障碍是它们的庞大规模。这项研究为调查这类网络提供了改进的工具。该方法也可用于三维空间中麦克斯韦方程的求解。本质上，我们必须计算非常大的矩阵的指数。这些矩阵太大，不允许使用标准软件。在本建议框架内开发的方法通过利用这些问题的固有结构来加快计算速度。工作的问题，这一建议需要在线性代数，正交多项式和有理函数，逼近理论和高斯型求积的背景。因此，该研究非常适合从事该项目的博士生。

This research project is concerned with the development and analysis of novel methods for the approximation of matrix functions and integrals defined by matrix functions. The research blends linear algebra and approximation theory. New rational Lanczos methods for Hermitian matrices with short recursion formulas are being developed. The existence of short recursion relations is of significant theoretical and practical interest. It speeds up the computations because each new rational Lanczos vector only has to be orthogonalized against a few of the most recently computed Lanczos vectors. The number of required explicit orthogonalizations is bounded independently of the number of rational Lanczos steps. The recursion relations define the orthogonal projection of a given matrix onto a rational Krylov subspace. The projection is represented by a pentadiagonal matrix, which also has a block structure. This matrix is analogous to the symmetric tridiagonal matrix associated with a (standard) Gauss quadrature rule, and is the basis for new algorithms for the evaluation of rational Gauss, Gauss-Radau rules, and Gauss-Lobatto quadrature rules. The research is an extension of the very important work by Gene Golub, and his collaborators, on matrices, moments, orthogonal polynomials, and quadrature. There may be connections to the structured matrices, such as the CMV-matrices, which arise in the context of orthogonal polynomials and orthogonal rational functions on the unit circle, as well as to to semi- and quasi-separable matrices. When the given matrix is non-Hermitian, rational Arnoldi and rational non-Hermitian Lanczos processes can be applied. The projections onto the rational Krylov subspace again have a structure, the exploitation of which is an important topic in linear algebra. The need to estimate matrix functionals arises in many applications, including the investigation of social networks and in Tikhonov regularization of ill-posed inverse problems. The principal investigator is studying developments of faster numerical methods for the evaluation, bounding or estimation of complicated nonlinear expressions that involve large symmetric or nonsymmetric matrices. The gain in speed is achieved by exploiting structure that until now has been ignored. The development of fast methods is important when solving large-scale problems of interest to scientists and engineers. These methods are applicable in the investigation of social networks, whose properties recently have received considerable attention, not only by scientists, but also by the New York Times. A major hurdle in the investigation of social networks is their large size. The research provides improved tools for investigating these kinds of networks. The methods can also be applied in solution methods for Maxwell's equation in three space-dimensions. Essentially, one has to compute the exponential of very large matrices. These matrices are much too large to allow the use of standard software. The methods to be developed within the framework of this proposal speed up the computations by exploiting inherent structure of these problems. Work on the problems of this proposal requires background in linear algebra, orthogonal polynomials and rational functions, approximation theory, and Gauss-type quadrature. Hence the research is well suited for doctoral students working on this project.