
Ublas : 
From: Gunter Winkler (gunter.winkler_at_[hidden])
Date: 20060412 04:21:01
On Tuesday 11 April 2006 20:27, asaf david wrote:
> hello
> i have a symetric and positive definite matrix A, and i'd like to find the
> second smallest eigenvector v of it (i believe the second smallest
> eignevector is the eigenvector corresponding for the second smallest
> eigenvalue, where the smallest is 0, but i'm not entirely sure about it..).
> Is there any straightforward way to do it in uBLAS? and if not, could you
> please recommend some other package?
You should definitely look for LAPACK or similar to compute eigenvalues and
eigenvectors. See http://www.netlib.org
In general you can compute eigenvectors and eigenvalues with a simple
iteration to compute the largest eigenvalue:
x = any nonzero start vector
iterate:
y = Ax
x = y / norm(y)
then norm(y)/norm(x) converges to the largest eigenvalue and x converges to
the corresponding eigenvector.
In order to compute the smallest eigenvalue of A you compute the largest
eigenvalue of the inverse of A. (You have to solve Ay=x in every iteration.)
If 0 is eigenvalue of A you have to compute the nullspace first and use a
start vector orthogonal to this nullspace and a more general solver, e.g.
solve norm(Ay  x) > inf, s.t. y is orthogonal to the nullspace.
All operations can be implemented using ublas, but a more specialized packages
will be more efficient.
mfg
Gunter