Eigenvalues of parameter-dependent quadratic eigenvalue problems form eigencurves. The critical points on these curves, where the derivative vanishes, are of practical interest. A particular example is found in the dispersion curves of elastic waveguides, where such points are called zero-group-velocity (ZGV) points. Recently, it was revealed that the problem of computing ZGV points can be modeled as a multiparameter eigenvalue problem (MEP), and several numerical methods were devised. Due to their complexity, these methods are feasible only for problems involving small matrices. In this paper, we improve the efficiency of these methods by exploiting the link to the Sylvester equation. This approach enables the computation of ZGV points for problems with much larger matrices, such as multi-layered plates and three-dimensional structures of complex cross-sections.

翻译：参数依赖二次特征值问题的特征值构成特征曲线。这些曲线上导数为零的临界点具有实际意义。一个典型例子出现在弹性波导的频散曲线中，此类点被称为零群速度点。近期研究表明，计算ZGV点的问题可建模为多参数特征值问题，并已发展出多种数值方法。由于计算复杂度较高，这些方法仅适用于小规模矩阵问题。本文通过利用与Sylvester方程的关联性，提升了现有方法的计算效率。该改进方案使得针对大规模矩阵问题（如多层平板结构和复杂截面三维结构）的ZGV点计算成为可能。