Rank-deficient matrices as a computational tool

✍ Scribed by W. Govaerts; B. Sijnave

• Publisher: John Wiley and Sons
• Year: 1997
• Tongue: English
• Weight: 129 KB
• Volume: 4
• Category: Article
• ISSN: 1070-5325
• DOI: 10.1002/(sici)1099-1506(199711/12)4:6<443::aid-nla118>3.0.co;2-d

✦ Synopsis

Rank-deficient matrices arise naturally in many applications. Detecting rank changes and computing parameter values for which a matrix has a prescribed (low) rank deficiency is a fundamental task in computing least squares and minimum norm solutions to systems of linear equations.

We describe an approach that originates from numerical continuation and bifurcation theory but has a wider applicability. It uses only linear solves with a bordered extension of the rank-deficient matrix and the transpose of that extension. We discuss the basic methods and their application in fundamental problems such as minimization and in more advanced problems in non-linear analysis. We present extensive numerical evidence in instructive test cases as well as in a chemical model (one-dimensional PDE) and a biological model (using the software package CONTENT for dynamical systems).