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

TL;DR

The work provides a unified framework for critical points of eigencurves in bivariate matrix pencils by introducing 2D points and ZGV points, which correspond to special eigenstructure where the derivative of the eigenvalue function vanishes. It connects these points to singular and regular two-parameter eigenvalue problems via operator determinants, and offers three numerical approaches: (i) a singular GEP-based method to obtain all 2D/ZGV points, (ii) a projected regular 2EP approach that enables efficient computation of subsets near targets, and (iii) a Gauss-Newton-type method with a fixed relative distance initialization (MFRD) for fast local convergence. The paper demonstrates the methods on diverse problems, including distance to instability, double eigenvalue problems, and two-parameter Sturm-Liouville and quadratic eigenvalue problems, and provides practical guidance and code for replication. Together, these results advance reliable global and local computation of critical points in multivariate matrix pencils with broad engineering and applied-mathematics relevance.

Abstract

We investigate critical points of eigencurves of bivariate matrix pencils $A+λB +μC$. Points $(λ,μ)$ for which $\det(A+λB+μC)=0$ form algebraic curves in $\mathbb C^2$ and we focus on points where $μ'(λ)=0$. 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 $(λ_0,μ_0)$, where $λ_0$ is a multiple eigenvalue of the pencil $(A+μ_0 C)+λB$, or, equivalently, there exist nonzero $x$ and $y$ such that $(A+λ_0 B+μ_0 C)x=0$, $y^H(A+λ_0 B+μ_0 C)=0$, 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.

Figures (11)

• Figure 1: Real eigencurves with ZGV (red) and 2D (white) points of Examples \ref{['exa:one']} (left) and \ref{['exa:three']} (right).
• Figure 2: Real eigenvalues and ZGV points of Example \ref{['ex:ex2x2']}.
• Figure 3: Convergence plot for Algorithm 4 applied to 2D points of Example \ref{['exa:one']}.
• Figure 4: Computational times (left hand side) and errors (right hand side) on random pencils.
• Figure 5: 2D-eigenvalues of Example \ref{['ex:lusu']}

Theorems & Definitions (36)

• Definition 1.1
• Theorem 1.2
• proof
• Definition 1.3
• Lemma 2.1
• Lemma 2.2
• proof
• Definition 2.3
• Lemma 2.4
• proof