Pseudospectra

Introduction

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.

Mathematical background

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, σ_ϵ(A), is the set of complex numbers λ such that

• λ is an eigenvalue of some matrix A+E, where the norm of the perturbation ‖E‖ < ϵ, or
• the resolvent norm ‖(A-λI)^(-1)‖ > 1/ϵ,

(the definitions are equivalent). This sense of "closeness" is trivial for Hermitian matrices, but interesting for others. 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 (from the Arpack package).

See the Pseudospectra gateway for details, references, and more.

Aside: the simple interface

To study a moderate-sized matrix with minimal user effort, follow this example:

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)

The figure shows a section of the complex plane with eigenvalues and contours of log10(ϵ).

Package context

Pseudospectra.jl (along with a QML-based GUI, in the forthcoming PseudospectraView package) is essentially a translation of the acclaimed MATLAB-based EigTool (homepage here), code now hosted on GitHub.

No endorsement or promotion of Pseudospectra.jl by the authors of EigTool is implied.

Specific documentation for Pseudospectra is a work in progress; a draft is available here. See the examples and tests for more.

Note on packaging/requirements

The plotting interface is somewhat schizophrenic. Drivers are included for Plots.jl and/or PyPlot.jl (i.e., PyPlot is a useful back end for Plots as used here; other Plots backends have been partially tested). Experimental support for Makie.jl is also available.

Although this package is designed with an eye to plotting results, the computational routines are usable without a plotting package, To avoid forcing potential users to install a particular one, none are specified in the formal package requirements. The setgs function can import one conditionally.

Some functions used for examples require other packages. They should give a useful complaint if invoked without that support.

Installation

This package should be included in the General registry by the time anyone else sees this paragraph, so the normal Pkg commands to add Pseudospectra should suffice.

In the interim, install by adding this repository explicitly with the package manager.

Pkg.add("https://github.com/RalphAS/Pseudospectra.jl")

Basic usage

Minimal use of the REPL interface is as follows:

using Plots
using Pseudospectra
A = your_matrix_generating_function()
ps_data = new_matrix(A)
driver!(ps_data)
# modify, e.g., for higher resolution
options = Dict{Symbol,Any}(:npts => 100)
driver!(ps_data,options)

This should show a contour plot of log10(ϵ) in the vicinity of the spectrum, which is the standard display of a spectral portrait. More elaborate capabilities are exhibited (as always) in the examples and test folders.

Disclaimer

This software is provided by the copyright holders and contributors "as is" and any express or implied warranties, including, but not limited to, the implied warranties of merchantability and fitness for a particular purpose are disclaimed. In no event shall the copyright holder or contributors be liable for any direct, indirect, incidental, special, exemplary, or consequential damages (including, but not limited to, procurement of substitute goods or services; loss of use, data, or profits; or business interruption) however caused and on any theory of liability, whether in contract, strict liability, or tort (including negligence or otherwise) arising in any way out of the use of this software, even if advised of the possibility of such damage.

Note on licensing

Most of the package is under a BSD license, in accordance with derivation from EigTool. See individual source files for exceptions.