Fande Kong <[email protected]> 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.
