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- W2947165005 abstract "Let $P : Omega subset {mathbb C} rightarrow {mathbb C}^{ntimes n}$ be given by $P(lambda) :=sum^m_{j=0}A_jphi_j(lambda),$ 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 $H : {mathbb C}^2 rightarrow {mathbb C}^{ntimes n}$ of the form $H(c, s) :=sum^m_{j=0}A_jpsi_j(c, s),$ 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." @default.
- W2947165005 created "2019-06-07" @default.
- W2947165005 creator A5017338530 @default.
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- W2947165005 date "2019-01-01" @default.
- W2947165005 modified "2023-10-18" @default.
- W2947165005 title "Sensitivity Analysis of Nonlinear Eigenproblems" @default.
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- W2947165005 doi "https://doi.org/10.1137/17m1153236" @default.
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