unsymmetric eigenvalue update

[Patrick Alken <patrick dot alken at colorado dot edu>]

Hi all,

  After reviewing the LAPACK and EISPACK eigenvalue routines,
I've changed my convergence criteria to declare failure after
30*N iterations. My original number of 30 was based on the
Numerical Recipes algorithm, which I now see was copied
from the EISPACK algorithm but with a typo of 30 instead of
30*N.

  I have run my code through Brian Gough's test program
on 12x12 and 200x200 random matrices, as well as some integer
matrix cases and find no significant disagreement with LAPACK.

  For those who expressed concern earlier about speed issues,
I'd like to share some things I found while studying this
problem:

1) In 1962 the Francis double-shift QR algorithm was published
   which typically converges in about 8n^3 flops if you only
   want eigenvalues. This is the algorithm which was implemented
   in EISPACK (hqr.f) and is also implemented in LAPACK as
   DLAHQR. Also this is the algorithm discussed in Golub &
   Van Loan and is presented in Numerical Recipes (which is just
   the EISPACK hqr routine converted to C).

2) In 1989, Bai and Demmel published their multi-shift QR
   algorithm and gave some numerical tests showing that it
   is generally faster than hqr from EISPACK. However they
   say its too complicated to derive an approximate number of
   flops required for the general problem. Furthermore,
   the multi-shift algorithm needs to call the double-shift
   algorithm to calculate the shifts. So it is impossible
   to dispense with the double-shift algorithm. This algorithm
   is implemented in LAPACK as DGEES (DGEES calls DLAHQR to
   get the shifts)

3) In a further significant development, in 2001 a multishift
   QR algorithm was proposed in the papers

   K. Braman et al, "The Multishift QR Algorithm: Part I:
   Maintaining Well Focused Shifts"

   and

   K. Braman et al, "The Multishift QR Algorithm: Part II:
   Aggressive Early Deflation"

   which introduces a way to deflate eigenvalues much more
   quickly than the standard test used in the previous two
   methods. The numerical results in these papers are very
   impressive and show convergence with far less flops than
   the LAPACK algorithm. Subsequent papers have extended this
   method to the QZ algorithm of the generalized eigenvalue
   problem, and as far as I can tell, this aggressive deflation
   method is considered the best QR algorithm currently
   available. If any experts are reading this please correct
   me!

So, the code in my gsl patch is equivalent to the HQR routine
in EISPACK, which has perfectly fine accuracy but is not
the fastest converging. I believe ultimately, the goal should
be to implement method #3 which will perform better than
LAPACK's DGEES, but is fairly complex and will take me some
time to learn/implement. Furthermore, the implementation of
methods 2 or 3 require the double-shift method 1 anyway, so
it was not a waste of time to code up the double-shift method.

I propose adding the double-shift method to gsl (assuming it
passes any further tests needed) which would be a perfectly
fine eigenvalue solver (EISPACK used it for a number of years)
until I (or someone else) can find the time to implement method 3.

Attached is the latest (final? :)) patch.

Patrick Alken

Attachment: gsl.unsymm.patch
Description: Text document