  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 $\mu_{0}$.
