Re: Symmetric matrices only!

[Adam Beneschan <adam@irvine.com>]

On Apr 17, 6:16 am, amado.al...@gmail.com wrote:
> The eigenvector solvers in Ada.Numerics.Generic_Real_Arrays require
> *symmetric* matrices! This is extremely silly. The world is full of
> nonsymmetric matrices. I've got a bunch of them to solve. Some very
> large, e.g. 1000x1000. Suggestions welcome. Thanks a lot.

We're in an area where I have no mathematical knowledge.  However, I
can quote something from AI95-296, which might explain why an
"extremely silly" decision was made:

'We considered providing subprograms for the determination of
eigenvalues and
eigenvectors of general real and complex matrices. Such matrices can
have
complex eigenvalues and therefore provision for these would have to be
in the
complex package. However, there are mathematical difficulties with
these general
cases which are in strong contrast to the real symmetric and Hermitian
matrices.
Thus, Numerical Recipes by Press, Flannery, Teukolsky and Vetterling
says
regarding the real case:

"The algorithms for symmetric matrices ... are highly satisfactory in
practice.
By contrast, it is impossible to design equally satisfactory
algorithms for the
nonsymmetric case. There are two reasons for this. First, the
eigenvalues of a
nonsymmetric matrix can be very sensitive to small changes in the
matrix
elements. Second, the matrix itself can be defective so that there is
no
complete set of eigenvectors. We emphasize that these difficulties are
intrinsic
properties of certain nonsymmetric matrices, and no numerical
procedure can cure
them."'

In the discussion section of the AI there's this, from John Barnes:

"Moreover, the eigenvalues and vectors of
nonsymmetric, non-Hermitian matrices have been removed because of
potential
computational difficulties."

I have no idea what all this means, but it doesn't sound silly to me.

                                     -- Adam