A numerical algorithm is proposed to explore in a systematic way the trajectories of the eigenvalues of non-Hermitian matrices in the parametric space and exploit this in order to find the locations of defective eigenvalues in the complex plane. These non-Hermitian degeneracies also called exceptional points (EP) have raised considerable attention in the scientific community as these can have a great impact in a variety of physical problems.

The method requires the computation of successive derivatives of two selected eigenvalues with respect to the parameter so that, after recombination, regular functions can be constructed. This algebraic manipulation permits the localization of exceptional points (EP), using standard root-finding algorithms and the computation of the associated Puiseux series up to an arbitrary order. This representation, which is associated with the topological structure of Riemann surfaces allows to efficiently approximate the selected pair in a certain neighbourhood of the EP.

Practical applications dealing with guided acoustic waves propagating in straight ducts with absorbing walls and in periodic guiding structures are given to illustrate the versatility of the proposed method and its ability to handle large size matrices arising from finite element discretization techniques. The fact that EPs are associated with optimal dissipative treatments in the sense that they should provide best modal attenuation is also discussed.

提出了一种数值算法，以系统的方式探索参数空间中非Hermitian矩阵的特征值的轨迹，并加以利用，以寻找复杂平面中有缺陷的特征值的位置。这些非Hermitian简并也被称为例外点（EP）在科学界引起了相当大的关注，因为它们会对各种物理问题产生重大影响。

该方法需要计算相对于参数的两个选定特征值的连续导数，以便在重组后可以构造规则函数。这种代数运算允许使用标准的求根算法以及任意阶数的相关Puiseux级数的计算来定位例外点（EP）。与Riemann表面的拓扑结构相关的这种表示允许有效地近似EP的某个邻域中的所选对。

给出了处理在带有吸收壁的直管中以及在周期性引导结构中传播的引导声波的实际应用，以说明所提出方法的多功能性及其处理有限元离散化技术产生的大尺寸矩阵的能力。从EP应该提供最佳模态衰减的意义上说，EP与最佳耗散处理相关联的事实也得到了讨论。