Eigenvalues
Here is the datafile
Arnoldi's method is a standard algorithm for computing a few
eigenvalues and corresponding eigenvectors for a large, sparse and
non-Hermitean matrix, A. The
algorithm is iterative and in iteration k there are k approximate
eigenvalues, but only a few of these are good approximations. k is
typically much less than the dimension, n, of the matrix. If n = 5000,
k is maybe 50, and the number of good approximations is perhaps 5. A
by-product of the algorithm is the residual norm, || A x
- λ x || / ||
x ||, of an approximate eigenpair (λ, x) (λ is the approximate
eigenvalue and x an
approximation of the eigenvector). This norm can be used to extract the
good approximations and one would continue iterating until one has the
number of required approximations.
Suppose you have never heard of the algorithm One way to understand
part of its behaviour is to compare the eigenvalues computed by the
algorithm with the exact eigenvalues.
The file arnoldi.mat contains two
arrays, one, ex, contains the exact eigenvalues and the
other, app, contains the approximations. I have not
supplied any residual norms, so you can use the distance between an
exact and an approximate eigenvalue as an estimate of the error.
We want to have answers to the following questions. The answers
should be presented using computer graphics (of course).
- How accurate are the approximations? Note that some
approximations are better than others by several orders of magnitude.
We would like to see this
difference in quality. We are interested in the small errors, the large
ones correspond to bad approximations and are of no interest.
- In what part of A's
spectrum are the good approximations located?
- We want to see the location of the eigenvalues in the complex plane together with errors in one plot.
Note that the eigenvalues are complex.
