  Let $T : \Omega \rightarrow \bbC^{n \times n}$ be a matrix-valued function
that is analytic on some simply-connected domain $\Omega \subset \bbC$. A point
$\lambda \in \Omega$ is an eigenvalue if the matrix $T(\lambda)$ is singular.
In this paper, we describe new localization results for nonlinear eigenvalue
problems that generalize Gershgorin's theorem, pseudospectral inclusion
theorems, and the Bauer-Fike theorem. We use our results to analyze three
nonlinear eigenvalue problems: an example from delay differential equations, a
problem due to Hadeler, and a quantum resonance computation.
