We discuss two approaches to solving the parametric (or stochastic) eigenvalue problem. One of them uses a Taylor expansion and the other a Chebyshev expansion. The parametric eigenvalue problem assumes that the matrix $A$ depends on a parameter $\mu$, where $\mu$ might be a random variable. Consequently, the eigenvalues and eigenvectors are also functions of $\mu$. We compute a Taylor approximation of these functions about $\mu_{0}$ by iteratively computing the Taylor coefficients. The complexity of this approach is $O(n^{3})$ for all eigenpairs, if the derivatives of $A(\mu)$ at $\mu_{0}$ are given. The Chebyshev expansion works similarly. We first find an initial approximation iteratively which we then refine with Newton's method. This second method is more expensive but provides a good approximation over the whole interval of the expansion instead around a single point. We present numerical experiments confirming the complexity and demonstrating that the approaches are capable of tracking eigenvalues at intersection points. Further experiments shed light on the limitations of the Taylor expansion approach with respect to the distance from the expansion point $\mu_{0}$.

翻译：本文探讨了两种求解参数化（或随机）特征值问题的方法。其中一种采用泰勒展开，另一种采用切比雪夫展开。参数化特征值问题假定矩阵 $A$ 依赖于参数 $\mu$，其中 $\mu$ 可能为随机变量。因此，特征值和特征向量也是 $\mu$ 的函数。我们通过迭代计算泰勒系数，构建这些函数关于 $\mu_{0}$ 的泰勒近似解。若已知 $A(\mu)$ 在 $\mu_{0}$ 处的导数，该方法求解所有特征对的复杂度为 $O(n^{3})$。切比雪夫展开原理类似：首先通过迭代获得初始近似解，再利用牛顿法进行精化。第二种方法计算量更大，但能在整个展开区间（而非单点附近）提供良好的近似。我们通过数值实验验证了复杂度分析，并证明这两种方法均可追踪交点处的特征值。进一步实验揭示了泰勒展开方法在展开点 $\mu_{0}$ 邻域外的局限性。