There is more to the computation of pseudospectra than the obvious algorithm of computing singular value decompositions on a grid and sending the results to a contour plotter. Other methods may be hundreds of times faster. The state of the art is reviewed, with emphasis on methods for dense matrices, and a MATLAB code is given.

/pdf/computation-of-pseudospectra-3wnm6cvsca.pdf

Computation of pseudospectra

Pseudospectra associated with the standard and generalized eigenvalue problems have been widely investigated in recent years. We extend the usual definitions in two respects, by treating the polynomial eigenvalue problem and by allowing structured perturbations of a type arising in control theory. We explore connections between structured pseudospectra, structured backward errors, and structured stability radii. Two main approaches for computing pseudospectra are described. One is based on a transfer function and employs a generalized Schur decomposition of the companion form pencil. The other, specific to quadratic polynomials, finds a solvent of the associated quadratic matrix equation and thereby factorizes the quadratic $\lambda$-matrix. Possible approaches for large, sparse problems are also outlined. A collection of examples from vibrating systems, control theory, acoustics, and fluid mechanics is given to illustrate the techniques.

/pdf/structured-pseudospectra-for-polynomial-eigenvalue-problems-3lt4f41cjd.pdf

Structured Pseudospectra for Polynomial Eigenvalue Problems, with Applications

A description is given of methods that have been used to analyze the spectrum of non-self-adjoint differential operators, emphasizing the differences from the self-adjoint theory. It transpires that even in cases in which the eigenfunctions can be determined explicitly, they often do not form a basis; this is closely related to a high degree of instability of the eigenvalues under small perturbations of the operator.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0024609302001248

Non-Self-Adjoint Differential Operators

http://www.ma.man.ac.uk/~ftisseur/reports/hiti02a.pdf

More on pseudospectra for polynomial eigenvalue problems and applications in control theory

Summary

1
An important task in applied population ecology is to understand how changes to individual life-history parameters, such as survival, growth and fecundity, affect population dynamics. Parameter changes, or perturbations, may be caused by deliberate attempts to manage populations (e.g. in pest control, harvesting or conservation) or they may be side-effects of pollution, genetic modification and climate change.

2
For organisms with complicated life cycles, links between individual life histories and population dynamics are made using population projection matrix (PPM) modelling.

3
Changes to individual, or groups of, life-history transition rates within a PPM have a nonlinear impact on the resulting eigenvalues. Conventional sensitivity analysis calculates the derivative of the perturbation-eigenvalue curve to provide tangential linear extrapolation. Until now, only the simulation of perturbed PPMs has captured nonlinear perturbation effects.

4
Here we describe the transfer function of a matrix perturbation. The transfer function captures analytically the true relationship between perturbation magnitude and PPM eigenvalues. This analytical link extends easily to multi-transition and multiple perturbations, promotes an understanding of matrix properties, and provides a simple method to predict the perturbation required to achieve a desired population rate of increase.

5
We use the transfer function approach to analyse a PPM for the desert tortoise Gopherus agasizzii Cooper, in the context of conservation management decisions.

6
Synthesis and applications. The transfer function offers a novel and powerful framework for the analysis of population projection matrices (PPMs), giving precise predictive power and analytical understanding of population-level responses to life-history perturbations, for example in the design of conservation, pest control and population harvesting strategies, prediction of population effects of pollution in ecotoxicology, and in ecological risk assessment. A useful focus is to set a target for the desired rate of increase (or decline) of a population, and use the transfer function to determine how best to achieve this rate.

B. Kelb

Papers

Spectral value sets: a graphical tool for robustness analysis