### Summary

Let $T : \Omega \rightarrow \mathbb{C}^{n \times n}$ be a matrix-valued function that is analytic on some simply-connected domain $\Omega \subset \mathbb{C}$. We call a point $\lambda \in \Omega$ an eigenvalue if the matrix $T(\lambda)$ is singular. Nonlinear eigenvalue problems arise in many applications, often from applying transform methods to analyze differential and difference equations. We describe new localization results for nonlinear eigenvalues that generalize Gershgorin’s theorem, pseudospectral inclusion theorems, the Bauer-Fike theorem, and others.

### Papers

A. Hood and D. Bindel, “Pseudospectral bounds on transient growth for higher order and constant delay differential equations,” Nov. 2016. Submitted to SIAM Journal on Matrix Analysis and Applictions.
@techreport{2016-transient-tr,
author = {Hood, Amanda and Bindel, David},
title = {Pseudospectral bounds on transient growth for higher order and constant delay differential equations},
month = nov,
year = {2016},
arxiv = {1611.05130},
status = {unrefereed},
submit = {Submitted to SIAM Journal on Matrix Analysis and Applictions.}
}

D. Bindel and A. Hood, “Localization Theorems for Nonlinear Eigenvalues,” SIAM Review, vol. 57, no. 4, pp. 585–607, Dec. 2015.
SIGEST feature article.
@article{2015-sirev,
author = {Bindel, David and Hood, Amanda},
title = {Localization Theorems for Nonlinear Eigenvalues},
journal = {SIAM Review},
publisher = {SIAM},
volume = {57},
number = {4},
pages = {585--607},
month = dec,
year = {2015},
notable = {SIGEST feature article.},
doi = {10.1137/15M1026511}
}


#### Abstract:

Let $T : \Omega \rightarrow {\Bbb C}^{n\times n}$ be a matrix-valued function that is analytic on some simply-connected domain $\Omega \subset {\Bbb C}$. 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.

D. Bindel and A. Hood, “Localization Theorems for Nonlinear Eigenvalues,” SIAM Journal on Matrix Analysis, vol. 34, no. 4, pp. 1728–1749, 2013.
2015 SIAG/LA award (best journal paper in applied LA in three years).
@article{2013-simax,
author = {Bindel, David and Hood, Amanda},
title = {Localization Theorems for Nonlinear Eigenvalues},
journal = {SIAM Journal on Matrix Analysis},
volume = {34},
number = {4},
pages = {1728--1749},
year = {2013},
doi = {10.1137/130913651},
arxiv = {http://arxiv.org/abs/1303.4668},
notable = {2015 SIAG/LA award (best journal paper in applied LA in three years).}
}


#### Abstract:

Let $T : \Omega \rightarrow {\Bbb C}^{n \times n}$ be a matrix-valued function that is analytic on some simply-connected domain $\Omega \subset {\Bbb C}$. 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.

### Talks

#### Dynamics via Nonlinear Pseudospectra

Foundations of Computational Math 2017
nep eigenboundsminisymposium external invited

#### Nonlinear Eigenvalue Problems: Theory and Applications

University of Arizona Math Colloquium
nep eigenboundscolloquium external invited

#### Nonlinear Eigenvalue Problems: Theory and Applications

Cornell Applied Math Colloquium
nep eigenboundscolloquium local

#### Localizing Nonlinear Eigenvalue Problems: Theory and Applications

SIAM LA 2015 (Prize Lecture)
nep eigenboundsmeeting external invited plenary

#### Eigenvalue Localization and Applications

NEP14 Workshop
eigenbounds nepmeeting external invited

#### Some perturbation theorems for nonlinear eigenvalue problems

Workshop on Dissipative Spectral Theory, Cardiff
eigenbounds matscat nep pml resonancemeeting external invited

#### Numerical Analysis of Resonances

Weyl at 100 Workshop (Fields Institute)
eigenbounds matscat nep pml resonancemeeting external invited

#### Analyzing Resonances via Nonlinear Eigenvalues

ICIAM
eigenbounds matscat nep resonanceminisymposium external invited

#### Resonances: Interpretation, Computation, and Perturbation

Cornell SCAN Seminar
eigenbounds matscat nep resonanceseminar local

#### Resonances: Interpretation, Computation, and Perturbation

Workshop in honor of Pete Stewart at UT Austin
eigenbounds matscat nep resonancemeeting external invited

#### Applications and Analysis of Nonlinear Eigenvalue Problems

Simon Fraser University NA Seminar
eigenbounds matscat nep resonance pmlseminar external invited

#### Resonances and Nonlinear Eigenvalue Problems

NYCAM
eigenbounds matscat nep resonancemeeting external

#### Numerical Analysis for Nonlinear Eigenvalue Problems

Cornell SCAN Seminar
eigenbounds matscat nep pml resonanceseminar local

#### Bounds and Error Estimates for Resonance Problems

SIAM Annual Meeting
eigenbounds matscat nep pml resonanceminisymposium external invited

#### Numerical Methods for Resonance Calculations

MSRI Workshop on Resonances
eigenbounds matscat nep resonancemeeting external invited

#### Bounds and Error Estimates for Nonlinear Eigenvalue Problems

Berkeley Applied Math Seminar
eigenbounds matscat nep pml resonanceseminar external invited

#### Error Bounds and Error Estimates for Nonlinear Eigenvalue Problems

Householder Symposium
eigenbounds nepminisymposium external invited

#### Spectral Inclusion Regions for Bifurcation Analysis

NYU NA Seminar
eigenboundsseminar local

#### Spectral Inclusion Regions for Bifurcation Analysis

Stanford NA Seminar
eigenboundsseminar external invited

#### Inclusion Regions for Bifurcation Analysis

UC Berkeley LAPACK Seminar
cis eigenboundsseminar local