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 $(A+\lambda_0 B+\mu_0 C)x=0$, $y^H(A+\lambda_0 B+\mu_0 C)=0$, 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.

翻译：本文研究双变量矩阵束 $A+\lambda B +\mu C$ 特征曲线的临界点。满足 $\det(A+\lambda B+\mu C)=0$ 的点 $(\lambda,\mu)$ 在 $\mathbb C^2$ 中构成代数曲线，我们重点关注满足 $\mu'(\lambda)=0$ 的点。根据工程应用中的术语，此类点被称为零群速度点。我们为零群速度点建立了普适理论，并证明它们构成二维点 $(\lambda_0,\mu_0)$ 的一个子集（在一般情形下取等号），其中 $\lambda_0$ 是矩阵束 $(A+\mu_0 C)+\lambda B$ 的多重特征值，或者等价地，存在非零向量 $x$ 和 $y$ 使得 $(A+\lambda_0 B+\mu_0 C)x=0$，$y^H(A+\lambda_0 B+\mu_0 C)=0$，且 $y^HBx=0$。我们提出了三种计算二维点与零群速度点的数值方法。第一种方法通过求解相关奇异双参数特征值问题的特征值来计算所有二维点（零群速度点）。第二种方法采用投影正则双参数特征值问题，可计算全部特征值或仅计算靠近给定目标的特征值子集。第三种方法是局部收敛的高斯-牛顿型方法，可从初始近似值计算单个二维点，而通过 Jarlebring、Kvaal 和 Michiels 提出的固定相对距离法可为所有二维点提供初始近似。在数值算例中，我们运用这些方法计算二维特征值、求解双重特征值问题、确定参数依赖二次特征值问题的零群速度点、评估稳定矩阵到失稳边界的距离，并寻找双参数 Sturm-Liouville 问题特征曲线的临界点。