Numerical methods for rectangular multiparameter eigenvalue problems, with applications to finding optimal ARMA and LTI models

Hochstenbach, Michiel E. (author), Košir, Tomaž (author), Plestenjak, Bor (author)

Standard multiparameter eigenvalue problems (MEPs) are systems of $k \ge 2$ linear $k$-parameter square matrix pencils. Recently, a new form of multipa-rameter eigenvalue problems has emerged: a rectangular MEP (RMEP) with only one multivariate rectangular matrix pencil, where we are looking for combinations of the parameters for which the rank of the pencil is not full. Applications include finding the optimal least squares autoregressive moving average (ARMA) model and the optimal least squares realization of autonomous linear time-invariant (LTI) dynamical system. For linear and polynomial RMEPs, we give the number of solutions and show how these problems can be solved numerically by a transformation into a standard MEP. For the transformation we provide new linearizations for quadratic multivariate matrix polynomials with a specific structure of monomials and consider mixed systems of rectangular and square multivariate matrix polynomials. This numerical approach seems computationally considerably more attractive than the block Macaulay method, the only other currently available numerical method for polynomial RMEPs.

• Language: English
• Keywords: ARMA, block Macaulay matrix, LTI, rectangular multiparameter eigenvalue problem, standard multiparameter eigenvalue problem
• Work type: Article
• Typology: 1.01 - Original Scientific Article
• Organization: FMF - Faculty of Mathematics and Physics
• Publication status: Published
• Publication version: Version of Record
• Year: 2024
• Number of pages: 23 str.
• Numbering: Vol. 31, iss. 2, art. e2540
• PID: 20.500.12556/RUL-154375
• UDC: 519.6
• ISSN on article: 1070-5325
• DOI: 10.1002/nla.2540
• COBISS.SI-ID: 172121859
• Publication date in RUL: 12.02.2024

Record is a part of a journal

• Title: Numerical linear algebra with applications
• Shortened title: Numer. linear algebra appl.
• Publisher: Wiley
• ISSN: 1070-5325
• COBISS.SI-ID: 17493849

Licences

• License: CC BY-NC-ND 4.0, Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International
• Link: http://creativecommons.org/licenses/by-nc-nd/4.0/
• Description: The most restrictive Creative Commons license. This only allows people to download and share the work for no commercial gain and for no other purposes.

Secondary language

• Language: Slovenian
• Keywords: ARMA, bločna Macaulayeva matrika, pravokotni večparametrični problem lastnih vrednosti, standardni večparametrični problem lastnih vrednosti

Projects

• Funder: ARRS - Slovenian Research Agency
• Project number: N1-0154
• Name: Verjetnostne metode za skupne in singularne probleme lastnih vrednosti