Solving the Parametric Eigenvalue Problem by Taylor Series and Chebyshev Expansion
Thomas Mach, Melina A. Freitag · SIAM Journal on Matrix Analysis and Applications · 2025
Abstract. 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 [Formula: see text] depends on a parameter [Formula: see text], where [Formula: see text] might be a random variable. Consequently, the eigenvalues and eigenvectors are also functions of [Formula: see text]. We compute a Taylor approximation of these functions about [Formula: see text] by iteratively computing the Taylor coefficients. The complexity of this approach is [Formula: see text] for all eigenpairs if the derivatives of [Formula: see text] at [Formula: see text] 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 of 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 [Formula: see text].