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**

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