Interpretation of the pseudospectra

The initial pseudospectra will be computed when the GUI starts up. This plot may itself be enough to indicate whether this matrix is far enough from normal for non-normality to be a potential concern. The first image below shows a matrix which is only mildly non-normal. The epsilon-pseudospectral boundaries are at a distance of about epsilon from the spectrum. The non-normality is probably not important.

The second image shows a matrix which is more significantly non-normal. Here, the epsilon-psuedospectra lie at a distance much greater than epsilon from the spectrum, and the non-normality would probably be important in a physical application.

For highly non-normal problems, the eigenvalue estimates returned by the Arnoldi iteration may in fact be incorrect (see the second example), even if they converge. Perhaps the more serious case is when the Ritz values returned are accurate approximations to the eigenvalues, but the eigenvalues are not physically significant due to the non-normality (see the third example).

When pronounced non-normality is seen in the initial plot created by the GUI, one thing to do is to use a finer grid for the pseudospectra computation. This will take longer, but may indicate the degree of non-normality more effectively. If the non-normality is significant, further computations should be performed before taking the eigenvalues returned at face value.

For more discussion see the

Pseudospectra GUI home page.