Linear algebra algorithms as dynamical systems

Abstract

Any logical procedure that is used to reason or infer either deductively or inductively so as to draw conclusions or make decisions can be called, in a broad sense, a realization process. A realization process usually assumes the recursive form that one state gets developed into another state by following a certain specific rule. Such an action is qualified as what is generally known as a dynamical system. In mathematics, especially for existence questions, a realization process often appears in the form of an iterative procedure or a differential equation. For years researchers have taken great effort to describe, analyze, and modify realization processes for various applications.The thrust in this exposition is to exploit the notion of dynamical systems as a special realization process for problems arising from the field of linear algebra. Several differential equations whose solutions evolve in some submanifolds of matrices are cast in fairly general frameworks of which special cases have been found to afford unified and fundamental insights into the structure and behavior of existing discrete methods and, now and then, suggest new and improved numerical methods. In some cases, there are remarkable connections between smooth flows and discrete numerical algorithms. In other cases, the flow approach seems advantageous in tackling very difficult open problems. Various aspects of the recent development and application in this direction are discussed in this paper.

Riemannian inexact Newton method for structured inverse eigenvalue and singular value problems

“…The problem of optimizing a function on a matrix manifold has received much attention in the scientific and engineering fields due to its peculiarity and capacity. Its applications include, but are not limited to, the study of eigenvalue problems [12,13,7,1,2,14,10,50,52,48,46,51], matrix low rank approximation [4,27], and nonlinear matrix equations [44,11]. Numerical methods for solving problems involving matrix manifolds rely on interdisciplinary inputs from differential geometry, optimization theory, and gradient flows.…”

Short note: An integrable numerical algorithm for computing eigenvalues of a specially structured matrix

“…[1][2][3][4] and references therein). For example, the -algorithm is nothing but the discrete KdV equation [5].…”

Quantization in Control Systems and Forward Error Analysis of Iterative Numerical Algorithms

“…However, in recent years, interest has been shown in this application of control theory, e.g. see Bhaya and Kaszkurewicz (2007); Chehab and Laminie (2005); Chu (2008). Most of this work is focused on design of new iterative methods and analysis of algorithms in exact precision.…”