Computing nearby non-trivial Smith forms. (January 2021)

• Record Type: Journal Article
• Authors: Giesbrecht, Mark; Haraldson, Joseph; Labahn, George
• Is Part Of: Journal of symbolic computation. Volume 102(2020)
• Page Start: 304
• Page End: 327
• Publication Date: 2021-01
• Subjects: Matrix polynomials -- Smith normal form -- Symbolic-numeric computing; Mathematics -- Data processing -- Periodicals; Numerical analysis -- Data processing -- Periodicals; Automatic programming (Computer science) -- Periodicals; Mathématiques -- Informatique -- Périodiques; Analyse numérique -- Informatique -- Périodiques; Programmation automatique -- Périodiques; Automatic programming (Computer science); Mathematics -- Data processing; Numerical analysis -- Data processing; Periodicals; Electronic journals; 510.285
• DOI: 10.1016/j.jsc.2019.10.019
• Languages: English
• ISSNs: 0747-7171

Abstract

Abstract: We consider the problem of computing the nearest matrix polynomial with a non-trivial Smith Normal Form. We show that computing the Smith form of a matrix polynomial is amenable to numeric computation as an optimization problem. Furthermore, we describe an effective optimization technique to find a nearby matrix polynomial with a non-trivial Smith form. The results are then generalized to include the computation of a matrix polynomial having a maximum specified number of ones in the Smith Form (i.e., with a maximum specified McCoy rank). We discuss the geometry and existence of solutions and how our results can be used for an error analysis. We develop an optimization-based approach and demonstrate an iterative numerical method for computing a nearby matrix polynomial with the desired spectral properties. We also describe an implementation of our algorithms and demonstrate the robustness with examples in Maple .