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We discuss two approaches to solving the parametric (or stochastic)eigenvalue problem. One of them uses a Taylor expansion and the other aChebyshev 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 $\mu$. Wecompute a Taylor approximation of these functions about $\mu_{0}$ byiteratively computing the Taylor coefficients. The complexity of this approachis $O(n^{3})$ for all eigenpairs, if the derivatives of $A(\mu)$ at $\mu_{0}$are given. The Chebyshev expansion works similarly. We first find an initialapproximation iteratively which we then refine with Newton's method. Thissecond method is more expensive but provides a good approximation over thewhole interval of the expansion instead around a single point.  We present numerical experiments confirming the complexity and demonstratingthat the approaches are capable of tracking eigenvalues at intersection points.Further experiments shed light on the limitations of the Taylor expansionapproach with respect to the distance from the expansion point $\mu_{0}$.

Solving the Parametric Eigenvalue Problem by Taylor Series and Chebyshev Expansion