In this example, the matrix A of size 
comes from an electromagnetism problem that deals with
the diffraction of a transverse-magnetic wave by a periodic 2D structure.
This example has been submitted by the Electromagnetism project at
CERFACS [8].
The electric field 
at time 
(where 
is the time
step), is computed using an iterative
scheme 
where 
tends to 0. Due to the
experimental assumptions, 
should tend to 0 as k increases.
Mathematically,
the scheme converges to 0 provided that the spectral radius of A is
smaller than 1. This fact holds in exact arithmetic.
But in finite precision the scheme diverges.
Figure 1 shows that the pseudospectrum
spreads out of the unit ball for relatively small matrix perturbations.
Figure 2 shows a zoom of the spectral portrait in the
neighbourhood of the point z=1: the contour line which is displayed
is the border of the 
-pseudospectrum of A for
.
Due to finite precision arithmetic, the iteration matrix is not
exactly A but a perturbed matrix 
. And indeed for relative
perturbations of the order of the machine double precision
(
), the 
-pseudospectrum includes complex
points of modulus larger than 1: these points can be eigenvalues
of the iteration matrix in finite precision.
This explains the possibility for the iterative scheme to diverge in
finite precision.
Figure 1: Spectral portrait of A
Figure 2: Zoom around the point z=1