Example matrix generators

landau_fox_li(N,F)

construct a Landau matrix of rank N for Fresnel number F

The discretization of a Fox-Li operator in the theory of laser cavities by ^[Landau1977] is one of the earliest applications of pseudospectra.

grcar(N)

construct a Grcar non-normal matrix of rank N

This is a popular example in the field of matrix iterations of a matrix whose spectrum is in the right half-plane but whose numerical range is not. It's also a popular example in the study of nonsymmetric Toeplitz matrices. The matrix was first described in ^[Grcar1989] and its pseudospectra were first plotted in ^{[Trefethen1991]}.

demmel(N,M)

construct Demmel's matrix of rank N with edge value M

Demmel's matrix was perhaps the first matrix whose pseudospectra (with N=3) appeared in ^[Demmel1987] Demmel devised the matrix in order to disprove a conjecture of Van Loan's ^{[VanLoan1985]}.

supg(N)

construct a SUPG matrix for an NxN mesh

The matrix arises from SUPG discretisation of an advection-diffusion operator.

This demo is based on a function by Mark Embree, which was essentially distilled from the IFISS software of Elman and Silvester. See ^{[Fischer1999]} and ^[DJS].

For this example, whilst the eigenvalues computed by eigs are inaccurate due to the sensitivity of the problem, the approximate pseudospectra are very good approximations to the true ones.

olmstead(N)

load Olmstead matrix from file (N ∈ {500,1000})

convdiff_fd(N)

construct matrix representation of 2-D NxN convection-diffusion operator

convdiff_fd(N) returns matrix C of dimension N^2 representing the operator -ϵ ∇^2 u + ∂u/∂y

on the unit square with Dirichlet boundary conditions. The discretisation is done using finite differences (see, e.g., ^{[Hemmingsson1998]}), resulting in large, sparse matrices. The code is based on a routine by Daniel Loghin.

advdiff(N,η=0.015)

construct the matrix representing an advection-diffusion operator via Chebyshev collocation.

orrsommerfeld(N,R=5722,α=1.0)

construct a matrix representing the Orr-Sommerfeld operator in the rank N+1 Chebyshev value-space basis. Note: this imposes a norm of dubious physical meaning, since a clever trick is used to suppress spurious eigenvalues.

^[HS1994] W.Z.Huang and D.M.Sloan, "The pseudospectral method for solving differential eigenvalue equations," J. Comput. Phys. 111, 399-409 (1994).

rdbrusselator(N)

load reaction-diffusion Brusselator matrix from file (N ∈ {800,3200})

trefethen_tutorial(N) -> B,A

compute the matrices used for tutorial purposes in ^{[Trefethen1999]}. A is the Chebyshev discretization of a Schrödinger operator with complex potential. B includes weight factors so that a basic L2 norm is appropriate.

kahan(N)

construct Kahan's matrix of rank N

Kahan's matrix ^[Kahan1966] was devised to illustrate that QR factorisation with column pivoting is not a fail-safe method for determining the rank of the matrix. Rank determination is related to the "distance to singularity" of a matrix ^[Higham1988], or in other words, how large a perturbation is needed to make the matrix singular. The pseudospectra are shown in ^{[Trefethen1991]}.