[petsc-users] Condition Number and GMRES iteration

[Jed Brown]

Fande Kong <fdkong.jd at gmail.com> writes:

> Thanks, Matt,
>
> It is a great paper. According to the paper, here is my understanding: for
> normal matrices, the eigenvalues of the matrix together with the
> initial residual completely determine the GMRES convergence rate. For
> non-normal matrices, eigenvalues are NOT the relevant quantities in
> determining the behavior of GMRES.
>
> What quantities we should look at for non-normal matrices? In other words,
> how do we know one matrix is easier than others to solve?  Possibly they
> are still open problems?!

You can use the pseudospectrum, but that isn't a convenient thing to
compute for large systems.

With respect to condition number: an orthogonal matrix is a normal
matrix of condition number 1 for which GMRES requires n iterations.