2010/3/23 Patrick Alken<address@hidden>:
On 03/23/2010 09:54 AM, Francesco Abbate wrote:
After looking into this a bit more, I found that the code is performing
balancing of matrices for the case of computing eigenvectors, which does not
preserve orthogonality of the Schur vectors. I probably did it this way
because its the LAPACK default. This issue should only affect the function
gsl_eigen_nonsymmv_Z(). The gsl_eigen_nonsymm_Z() function should give you
an orthogonal matrix Z, provided you haven't turned on balancing.
The only way to give the user control over balancing in the eigenvector case
would be to make another function like gsl_eigen_nonsymmv_params()...which
may be best solution.
Otherwise for now you can just use gsl_eigen_nonsymm_Z() to get the matrix
Z, and a separate call to gsl_eigen_nonsymmv() to get the eigenvectors. Or
if you really need something efficient right away, go to line 98 of
eigen/nonsymmv.c and change it to:
gsl_eigen_nonsymm_params(1, 0, w->nonsymm_workspace_p);
Ok, Patrick, that looks ok. Probably a small fix should be included in
the release of GSL as you suggests by adding a parameter that you can
configure also for the 'nonsymmv' function.
I'll think a little bit more about whether it would make sense to give the
user the option of balancing for the nonsymmv functions. Out of curiosity,
why do you need both the Schur vectors and the eigenvectors?
I'm developing a software, GSL shell, that offer an high level
interface to GSL routines. I was implementing the routines for solving
eigensystems and for completeness I've also provided a "shur" function
that return the Schur decomposition of the matrix. If someone want to
try out, the Subverson repository at Savannah is up to date.
Here a sample session:
--------------------------------------------------------------------------------------
GSL Shell, Copyright (C) 2009 Francesco Abbate
GNU Scientific Library, Copyright (C) The GSL Team
Lua 5.1.4, Copyright (C) 1994-2008 Lua.org, PUC-Rio (complex double int32)
dofile('examples/matrix-algebra.lua')
A = vandermonde {-1, -2, 3, 4}
=A
[ -1 1 -1 1 ]
[ -8 4 -2 1 ]
[ 27 9 3 1 ]
[ 64 16 4 1 ]
e, v = eignonsymmv(A)
=e
[ -6.41391 ]
[ 5.54555+3.08545i ]
[ 5.54555-3.08545i ]
[ 2.3228 ]
= cmul(cinverse(v), c(A), v)
[ -6.41391 0 0 0 ]
[ 0 5.54555+3.08545i 0 0 ]
[ 0 0 5.54555-3.08545i 0 ]
[ 0 0 0 2.3228 ]
T, Z = schur(A)
= mul(Z, T, inverse(Z))
[ -1 1 -1 1 ]
[ -8 4 -2 1 ]
[ 27 9 3 1 ]
[ 64 16 4 1 ]
----------------------------------------------------------
Francesco