20.500.12556/RUL-177757 On properties and numerical computation of critical points of eigencurves of bivariate matrix pencils 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. 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 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 true false true Angleški jezik Slovenski jezik Članek v reviji 2026-01-06 13:56:52 2026-01-06 13:57:00 2026-06-12 04:18:36 0000-00-00 00:00:00 2026 0 0 36 str. iss. 1, art. 6 Vol. 63 Mar. 2026 0000-00-00 Zaloznikova Objavljeno NiDoloceno 0000-00-00 0000-00-00 0000-00-00 519.6 0008-0624 10.1007/s10092-025-00677-6 263609859 s10092-025-00677-6.pdf s10092-025-00677-6.pdf 1 8E84EB56A2EE71FFDB769A4640F53E37 f74ab2ce2290cd27db9ea5dc22390ac27c4998d1ec4f292a5ec3866d360caa3b 1a3a08a8-659c-11f1-9b0d-0050569b8976 https://repozitorij.uni-lj.si/Dokument.php?lang=slv&id=235381 https://link.springer.com/article/10.1007/s10092-025-00677-6 1 31ab70ef-eaff-11f0-bf92-0050569b8976 https://repozitorij.uni-lj.si/Dokument.php?lang=slv&id=225049 Fakulteta za matematiko in fiziko 0 0 0