Pseudospectra.jl

Pseudospectra is a Julia package for computing pseudospectra of non-symmetric matrices, and plotting them along with eigenvalues ("spectral portraits"). Some related computations and plots are also provided.

Introduction

Whereas the spectrum of a matrix is the set of its eigenvalues, a pseudospectrum is the set of complex numbers "close" to the spectrum in some practical sense.

More precisely, the ϵ-pseudospectrum of a matrix A, $\sigma_{\epsilon}(A)$, is the set of complex numbers $\lambda$ such that

• $\lambda$ is an eigenvalue of some matrix $A+E$, where the perturbation $E$ is small: $\|E\| < \epsilon$
• the resolvent at $\lambda$ has a large norm: $\|(A-λI)^{-1}\| > 1/\epsilon$,

(the definitions are equivalent). Specifically, this package is currently limited to the unweighted 2-norm.

Among other things, pseudospectra:

• elucidate transient behavior hidden to eigen-analysis, and
• indicate the utility of eigenvalues extracted via iterative methods like eigs.

This package facilitates computation, display, and investigation of the pseudospectra of matrices and some other representations of linear operators.

Spectral portraits

It is customary to display pseudospectra as contour plots of the logarithm of the inverse of the resolvent norm $\epsilon = 1/\|(A-zI)^{-1}\|$ for $z$ in a subset of the complex plane. Thus $\sigma_{\epsilon}(A)$ is the union of the interiors of such contours. Such plots, sometimes called spectral portraits, are the most prominent product of this package.

The figure shows a section of the complex plane with eigenvalues and contours of log10(ϵ). It was generated by the following code:

using Plots, Pseudospectra, LinearAlgebra
n=150
B=diagm(1 => fill(2im,n-1), 2 => fill(-1,n-2), 3 => fill(2,n-3), -2 => fill(-4,n-2), -3 => fill(-2im, n-3))
spectralportrait(B)

Credit

Pseudospectra.jl is largely a translation of the acclaimed MATLAB-based EigTool (homepage here)

References

• The Pseudospectra gateway.
• L.N. Trefethen and M.Embree, Spectra and Pseudospectra; The Behavior of Nonnormal Matrices and Operators, Princeton 2005,