On properties and numerical computation of critical points of eigencurves of bivariate matrix pencils

Plestenjak, Bor (author)

We investigate critical points of eigencurves of bivariate matrix pencils $A+\lambda B +\mu C$. Points $(\lambda,\mu)$ for which $\det(A+\lambda B+\mu C)=0$ form algebraic curves in ${\mathbb C}^2$ and we focus on points where $\mu’(\lambda)=0$. Such points are referred to as zero-group-velocity (ZGV) points, following terminology from engineering applications. We provide a general theory for the ZGV points and show that they form a subset (with equality in the generic case) of the 2D points $(\lambda_0,\mu_0)$, where $\lambda_0$ is a multiple eigenvalue of the pencil $(A+\mu_0 C)+\lambda B$, or, equivalently, there exist nonzero $x$ and $y$ such that $(A+\lambda_0 B+\mu_0 C)x=0$, $y^H(A+\lambda_0B+\mu_0 C)=0$, and $y^HBx=0$. We introduce three numerical methods for computing 2D and ZGV points. The first method calculates all 2D (ZGV) points from the eigenvalues of a related singular two-parameter eigenvalue problem. The second method employs a projected regular two-parameter eigenvalue problem to compute either all eigenvalues or only a subset of eigenvalues close to a given target. The third approach is a locally convergent Gauss–Newton-type method that computes a single 2D point from an inital approximation, the later can be provided for all 2D points via the method of fixed relative distance by Jarlebring, Kvaal, and Michiels. In our numerical examples we use these methods to compute 2D-eigenvalues, solve double eigenvalue problems, determine ZGV points of a parameter-dependent quadratic eigenvalue problem, evaluate the distance to instability of a stable matrix, and find critical points of eigencurves of a two-parameter Sturm–Liouville problem.

• Language: English
• Keywords: zero-group-velocity point, 2D point, bivariate matrix pencil, singular two-parameter eigenvalue problem, 2D-eigenvalue, double eigenvalue, distance to instability, two-parameter Sturm-Liouville 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: 2026
• Number of pages: 36 str.
• Numbering: Vol. 63, iss. 1, art. 6
• PID: 20.500.12556/RUL-177757
• UDC: 519.6
• ISSN on article: 0008-0624
• DOI: 10.1007/s10092-025-00677-6
• COBISS.SI-ID: 263609859
• Publication date in RUL: 06.01.2026

Record is a part of a journal

• Title: Calcolo
• Shortened title: Calcolo
• Publisher: Springer Nature, Consiglio Nazionale delle Ricerche Istituto di Informatica e Telematica
• ISSN: 0008-0624
• COBISS.SI-ID: 513854233

Licences

• License: CC BY 4.0, Creative Commons Attribution 4.0 International
• Link: http://creativecommons.org/licenses/by/4.0/
• Description: This is the standard Creative Commons license that gives others maximum freedom to do what they want with the work as long as they credit the author.

Secondary language

• Language: Slovenian
• Keywords: točka ničelne skupinske hitrosti, 2D točka, bivariatni matrični šop, singularni dvoparametrični problem lastnih vrednosti, 2D lastna vrednost, dvojna lastna vrednost, razdalja do nestabilnosti, dvoparametrični Sturm-Liouvilleov problem

Projects

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

• Funder: ARIS - Slovenian Research and Innovation Agency
• Project number: P1-0294
• Name: Računsko intenzivne metode v teoretičnem računalništvu, diskretni matematiki, kombinatorični optimizaciji ter numerični analizi in algebri z uporabo v naravoslovju in družboslovju