Title: Randomized methods for computing joint eigenvalues, with applications to multiparameter eigenvalue problems and root finding Show Chinese title

Title: 计算联合特征值的随机方法，及其在多参数特征值问题和根查找中的应用

Abstract: It is well known that a family of $n\times n$ commuting matrices can be simultaneously triangularized by a unitary similarity transformation. The diagonal entries of the triangular matrices define the $n$ joint eigenvalues of the family. In this work, we consider the task of numerically computing approximations to such joint eigenvalues for a family of (nearly) commuting matrices. This task arises, for example, in solvers for multiparameter eigenvalue problems and systems of multivariate polynomials, which are our main motivations. We propose and analyze a simple approach that computes eigenvalues as one-sided or two-sided Rayleigh quotients from eigenvectors of a random linear combination of the matrices in the family. We provide some analysis and numerous numerical examples, showing that such randomized approaches can compute semisimple joint eigenvalues accurately and lead to improved performance of existing solvers. Show Chinese abstract

Abstract: 众所周知，一族互换的$n\times n$矩阵可以通过一个酉相似变换同时上三角化。 上三角矩阵的对角线元素定义了该族的$n$联合特征值。 在本工作中，我们考虑数值计算此类联合特征值的近似值的任务，针对一族（几乎）互换的矩阵。 该任务出现在多参数特征值问题和多元多项式系统的求解器中，这是我们的主要动机。 我们提出并分析了一种简单的方法，该方法从该族矩阵的随机线性组合的特征向量中计算特征值作为单侧或双侧瑞利商。 我们提供了一些分析和大量数值例子，表明这种随机化方法可以准确计算半单联合特征值，并提高现有求解器的性能。