Sensitivity Analysis of Nonlinear Eigenproblems
Rafikul Alam, Sk. Safique Ahmad · SIAM Journal on Matrix Analysis and Applications · 2019
Let $P : \Omega \subset {\mathbb C} \rightarrow {\mathbb C}^{n\times n}$ be given by $P(\lambda) :=\sum^m_{j=0}A_j\phi_j(\lambda),$ where $ \phi_j : \Omega \rightarrow {\mathbb C}$ for $j=0, 1, \ldots, m$ 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 $\mathrm{cond}(\lambda, P)$ of a simple eigenvalue $\lambda$ of $P.$ Specially, we present an eigenvector-free representation of $\mathrm{cond}(\lambda, P)$ 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 $H : {\mathbb C}^2 \rightarrow {\mathbb C}^{n\times n}$ of the form $H(c, s) :=\sum^m_{j=0}A_j\psi_j(c, s),$ where $ \psi_j : {\mathbb C}^2 \rightarrow {\mathbb C}$ for $ j= 0, 1, \ldots, m$ are homogeneous functions of degree $\ell.$ 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 $\mathrm{cond}((\lambda, \mu), H)$ of a simple eigenvalue $(\lambda, \mu)$ of $H.$ Our eigenvector-free representation of $\mathrm{cond}((\lambda, \mu), H)$ 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.