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
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