Nonlinear eigenvalue bounds

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.

Nonlinear eigenvalue bounds

Summary

Papers

Abstract:

Abstract:

Talks

Nonlinear Eigenvalue Localization for Damping Bounds

Dynamics via Nonlinear Pseudospectra

Nonlinear Eigenvalue Problems: Theory and Applications

Nonlinear Eigenvalue Problems: Theory and Applications

Localizing Nonlinear Eigenvalue Problems: Theory and Applications

Eigenvalue Localization and Applications

Some perturbation theorems for nonlinear eigenvalue problems

Numerical Analysis of Resonances

Analyzing Resonances via Nonlinear Eigenvalues

Resonances: Interpretation, Computation, and Perturbation

Resonances: Interpretation, Computation, and Perturbation

Applications and Analysis of Nonlinear Eigenvalue Problems

Resonances and Nonlinear Eigenvalue Problems

Numerical Analysis for Nonlinear Eigenvalue Problems

Bounds and Error Estimates for Resonance Problems

Numerical Methods for Resonance Calculations

Bounds and Error Estimates for Nonlinear Eigenvalue Problems

Error Bounds and Error Estimates for Nonlinear Eigenvalue Problems

Spectral Inclusion Regions for Bifurcation Analysis

Spectral Inclusion Regions for Bifurcation Analysis

Inclusion Regions for Bifurcation Analysis