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Re: Sparse Eigenvalues
Re: Sparse Eigenvalues
Sun, 21 Dec 2008 17:02:51 +0100
Mozilla-Thunderbird 184.108.40.206 (X11/20081018)
Chaman Singh Verma wrote:
Well, here is my sincere question. It is my observation that eigs (
based on ARPACK code) seems to take atleast "N" sparse-matrix-vector
which could be quite expensive for reasonably large matrices. One of
where someone may need second eigenvalue is for graph partitioning (
Google rank Matrix). For typical graph of 200,000 rows and 100-200
columns, it takes
more than 45-50 minutes on single node machine, which is not
competitive to other
graph partitioning methods such as METIS.
The google rank matrix is just a power method to find the largest
eigenvalue, and the google rank matrices is documented as converging in
about 6 iterations of the power method. Google page rank doesn't fing
the second largest eigenvalue at all. ARPACK is not the same beast!!
If you only want the largest eigenvalue then use the power method. What
ARPACK gets you is the N largest eigenvalues (or N smallest or N closest
to a particular value), and to do that it uses a method a bit like the
Lanzcos method with restarting.. Lanzcos method finds the largest
eigenvalue, then forms a new problem with the largest eigenvalue removed
and repeats.. The new problem is formulated with multiple matrix vector
products.. Arnoldi's method adds a restarting step such that it can
handle eigenvalues that are close together without convergence issues.
How did you formulate the problems to eigs? Perhaps there is an issue in
the manner eigs is calls. For example a function call for the matrix
vector product is likely to be slower than a call to blas xGEMM or the
sparse equivalent internal to eigs itself.
Frankly, I've never looked at how METIS does its graph partitioning, but
I didn't have the impression it used an eigenvalue method.. METIS has a
repudiation as one of the better graph partitioner out there however.
Is the matrix positive definite? Are there eigenvalues in the set you
are looking for that are close together? If the matrix is PD and you
don't have eigenvalues close together a straight Lanzcos method will
probably be the fastest.
Here is my query to all the knowledge people.
(1) What is the most competitive algorithms for finding few
compared to ARPACK. Has somebody done any study and have some
number to show its
Getting rid of the restarting in the Arnoldi technique will certainly
reduce the number, but at the cost of poor convergence if there are
eigenvalues close together.
(2) Can we reduce the number of Matrix-Vec Operations ?
(3) Can spectral decomposition beat METIS type decompositions ?
Perhaps, but I'm an engineer and not a mathematician, so you might want
to ask someone who knows more..
David Bateman address@hidden
35 rue Gambetta +33 1 46 04 02 18 (Home)
92100 Boulogne-Billancourt FRANCE +33 6 72 01 06 33 (Mob)
Re: Sparse Eigenvalues, Jaroslav Hajek, 2008/12/20