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},
  link = {http://arxiv.org/pdf/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

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