An example in Electromagnetism



next up previous
Next: Parallel implementation Up: Parallel Computation of Spectral Previous: Spectral portraits of

An example in Electromagnetism

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



next up previous
Next: Parallel implementation Up: Parallel Computation of Spectral Previous: Spectral portraits of

Contact: toumazou@cerfacs.fr
Last Update: