Solving the Parametric Eigenvalue Problem by Taylor Series and Chebyshev Expansion

TL;DR

This paper introduces two methods, Taylor and Chebyshev expansions, for solving parametric eigenvalue problems where matrix dependence on a parameter affects eigenvalues and eigenvectors, with numerical validation of their effectiveness and limitations.

Contribution

The paper presents novel iterative methods using Taylor and Chebyshev expansions to efficiently approximate parametric eigenvalues and eigenvectors, including complexity analysis and numerical validation.

Findings

Both methods have complexity O(n^3) for all eigenpairs.

The Chebyshev expansion provides a good approximation over an entire interval.

The Taylor expansion's accuracy diminishes with distance from the expansion point.

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 $A$ 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 $\mu_{0}$.