This is a question for a smarter way of numerical computation.
When I diagonalize a certain type of Vandermonde-matrices in Pari/GP ("mateigen(M)"), for instance of size 16x16 then this can be done in say a couple seconds and internal float precision of, say 200 digits (my default, it might be possible with less). If I want to use size 32x32 I need then say 800 or even 1200 digits internal float precision. The size 64x64 is currently my favorite extent; but it needs more than 3000 digits (I use 4000 to avoid error-abortions) and last time I've tried, was 6000 secs. Also - if something is going wrong, say too few internal digits or insufficient memory, then it may happen, that Pari/GP calculates 1 hour - but then detects some error ("insufficient eigen sapce" - means I need more precision) - and everything done is lost....
So I'd like to try other methods which might allow step-wise improvement in some fashion - partial computation with submatrices, increasing accuracy or the like. For instance I do not yet see a possibility to take advantage of the procedure to compute the roots of the characteristic polynomial in advance - there is no procedure to compute the eigenvectors which exploits the knowledge of the eigenvalues. So my question (in general, not dependend on Pari/GP- I might program this myself):
Q: Is there some procedure to split the diagonalization into more distinct steps whose results solve the problem cumulatively?