The first step when solving an infinite-dimensional eigenvalue problem is often to discretize it. We show that one must be extremely careful when discretizing nonlinear eigenvalue problems. Using examples, we show that discretization can: (1) introduce spurious eigenvalues, (2) entirely miss spectra, and (3) bring in severe ill-conditioning. While there are many eigensolvers for solving matrix nonlinear eigenvalue problems, we propose a solver for general holomorphic infinite-dimensional nonlinear eigenvalue problems that avoids discretization issues, which we prove is stable and converges. Moreover, we provide an algorithm that computes the problem's pseudospectra with explicit error control, allowing verification of computed spectra. The algorithm and numerical examples are publicly available in $\texttt{infNEP}$, which is a software package written in MATLAB.

翻译：求解无限维特征值问题的第一步通常是对其进行离散化。我们证明，在离散化非线性特征值问题时必须格外谨慎。通过实例，我们表明离散化可能导致：(1) 引入伪特征值，(2) 完全遗漏谱，以及(3) 引发严重的病态性。虽然已有多种求解矩阵非线性特征值问题的特征值求解器，但我们提出了一种适用于一般全纯无限维非线性特征值问题的求解器，该求解器能避免离散化问题，并且我们证明了其稳定性和收敛性。此外，我们提供了一种算法，可通过显式误差控制计算问题的伪谱，从而验证计算所得的谱。该算法和数值示例已在 $\texttt{infNEP}$ 中公开，这是一个用 MATLAB 编写的软件包。