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Re: a few eigenvalues

From: A S Hodel
Subject: Re: a few eigenvalues
Date: Fri, 16 Mar 2001 09:15:35 -0600


does anybody know how we can efficiently compute in octave only the
first k eigenvectors (with largest eigenvalues) of a symmetric matrix?



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)

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.

Octave is freely available under the terms of the GNU GPL.

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