FW: eigen-vector-value-problem

["Daniel Lozier" <lozier@nist.gov>]

Editor's Note:  I regret the delay in posting this message, and
hope it has caused no serious inconvenience. -- Dan Lozier

-----Original Message-----
From: Vadim Kuznetsov [mailto:vadim@amsta.leeds.ac.uk]
Sent: Thursday, March 08, 2001 10:53 AM
To: Dan Lozier
Subject: eigen-vector-value-problem

Dear Dan,
       I would like this question to be posted to the opsf list.
Thanks.
Regards,
                    Vadim

I want to ask for opinions about one `strange spectral problem'
where, in a sense, the spectrum is the same as the eigenvalues. 
Let us call this thing an `eigen-vector-value-problem'.

Technically, one solves the following system of n equations:

(*)         \sum_k  A_{ijk} x_k = x_i x_j,        i,j,k=1,...,n,
 
assuming that it is compatible. Compatibility can be rephrased
as the commutativity of the matrices A_{i..} and of the matrices
A_{.j.} collected from the tensor A_{ijk}. Anyway, it is assumed.

HOW TO SOLVE SUCH (QUADRATIC?) PROBLEMS EFFECTIVELY?
ON COMPUTER?

Notice that spectrum here is `the same' as eigenvalues, therefore
for a given A we have to find only the vector x.

(*) can be realized as either (inverted) linearization problem or,
when n=infinity and the operator A acts in some functional space,
as a product formula for a special function.

So, my question can also be changed to the following: How to use
product formulas for producing effective numerical methods for
calculating the special fuction itself?  After all (*) is a very
particular spectral problem and one (maybe) can use this fact to 
invent a fast numerical algorithm in order to calculate x. Notice
that x can be out of the hypergeometric class, so that a problem
of its calculation can be very non-trivial task.