Sensitivity analysis of nonlinear eigenproblems

Abstract

Let P : Ω ⊂ C → Cn×n be given by P(λ):= m j =0 Ajφj(λ), where φj : Ω → C for j = 0, 1, . . ., m are suitable functions. We present an eigenvector-free framework for the sensitivity analysis of eigenvalues of P. We analyze the Fréchet differentiability of a simple eigenvalue of P as a function of P and derive two equivalent representations of the Fréchet derivative and the gradient of the eigenvalue. Further, we derive three equivalent representations of the condition number cond(λ, P) of a simple eigenvalue λ of P. Specially, we present an eigenvector-free representation of cond(λ, P) which generalizes a result due to Smith [Numer. Math., 10 (1967), pp. 232–240] for a standard eigenvalue problem to the case of a nonlinear eigenvalue problem and provides an alternative viewpoint of the sensitivity of eigenvalues. In the second part, we consider a homogeneous matrix-valued function H : C2 → Cn×n of the form H(c, s):= m j =0 Ajψj(c, s), where ψj : C2 → C for j = 0, 1, . . ., m are homogeneous functions of degree . We present a simple and concise eigenvector-free framework for the sensitivity analysis of eigenvalues of H that avoids the apparatus of projective spaces. We analyze Fréchet differentiability of a simple eigenvalue of H as a function of H and derive two equivalent representations of the Fréchet derivative and the gradient of the eigenvalue. Furthermore, we derive three equivalent representations of the condition number cond((λ, μ), H) of a simple eigenvalue (λ, μ) of H. Our eigenvector-free representation of cond((λ, μ), H) generalizes Smith’s eigenvector-free representation of the condition number of a simple eigenvalue of a matrix to the case of a homogeneous nonlinear eigenproblem. © 2019 Society for Industrial and Applied Mathematics