unsymmetric eigenvalue problem

[Patrick.Alken@colorado.edu]

Hello all,

  I have recently needed to compute eigenvalues of unsymmetric
matrices, and so I wrote a code based in GSL to do it since
one does not currently exist. I followed the QR algorithm
outlined in Golub & Van Loan, chapter 7.

  I have attached a patch file to this email containing all
my code. Right now only the eigenvalue problem is solved,
the code will not compute eigenvectors. The patch is against
gsl-1.8 so just cd into a fresh gsl-1.8 and

patch -p0 < gsl.unsymm.patch

You will probably need to run automake again in the toplevel
dir to update the Makefile.am's. I have also attached a
test program eig.c to illustrate the use of the solver.

The patch creates 3 new source files:

linalg/hessenberg.c
eigen/balance.c
eigen/unsymm.c

and 1 header file
eigen/balance.h

hessenberg.c contains gsl_linalg_hessenberg() which reduces
a matrix to upper Hessenberg form.

balance.c contains balance_matrix() to do the standard matrix
balancing before finding eigenvalues.

unsymm.c contains the Francis QR algorithm to find the eigenvalues
of a general unsymmetric real matrix:
  gsl_eigen_unsymm_alloc()
  gsl_eigen_unsymm_free()
  gsl_eigen_unsymm() --- Solves A x = \lambda x

I am happy to contribute this to GSL if the developers want to
use it. Also, if there is interest, I would be willing to look
at coding up an eigenvector solver too for the unsymmetric
eigenvalue program - I only needed eigenvalues for my current
project so I haven't looked at the eigenvector problem.

Also if there is interest I'd be interested in looking at the
generalized eigenvalue problem since I feel GSL is incomplete
without an Ax = sBx solver.

Please test and give feedback.

Thanks,
Patrick Alken
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