Re: a few eigenvalues

[A S Hodel]

> Hi,
>
> does anybody know how we can efficiently compute in octave only the
> first k eigenvectors (with largest eigenvalues) of a symmetric matrix?
>
> thanks,
> Nikos
>
> --
> http://www.science.uva.nl/~vlassis

You may want to look at Parlett's book "The Symmetric Eigenvalue Problem."
The methods that I'm aware of involve a power iteration

Q_0 = (orthonormal n x k matrix)
while (not happy)
  W_n = A Q_n
  [Q_{n+1}, R_{n+1}] = qr(W)
endwhile

If A = A' > 0, then Q will converge to an orthonormal basis of the
span of the dominant k eigenvectors.  The leading eigenvectors converge
more quickly than the later ones, so it may be valuable to let
Q_n be n x k1 , k1 > k.

Alternatives include using the Arnoldi/Krylov iteration, which yield
Ritz pairs of approximate eigenvalues/eigenvectors that rapidly converge
to the extremal (largest, smallest) eigenvalues and associated eigenvectors.


Each of these methods is covered in Golub and Van Loan's book "Matrix
Computations," which is a must-read for anyone who wants to work with numerical
linear algebra.




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