[SciPy-User] Eigenvectors of sparse symmetric matrix
Lutz Maibaum
lutz.maibaum@gmail....
Mon Oct 25 15:35:21 CDT 2010
On Oct 25, 2010, at 12:02 PM, Pauli Virtanen wrote:
> Mon, 25 Oct 2010 11:14:16 -0700, Hao Xiong wrote:
>>> You can try playing with setting the maxiter parameter to allow ARPACK
>>> to spend more iterations on the problem.
>>
>> I tried maxiter=100000 and still got zero vectors. I must be missing
>> something.
>
> Can you perhaps write a small routine that generates such a fake matrix,
> or save it to a file and put one online. This might help in finding out
> why the routine doesn't behave as expected.
I think I am seeing the same problem as Hao. I have uploaded a sample matrix exhibiting this problem:
http://stanford.edu/~maibaum/matrix.mtx.gz
A simple test to illustate the problem is the following:
In [1]: from scipy.io import mmread
In [2]: import scipy.sparse.linalg.eigen
In [3]: import numpy as np
In [4]: a = mmread("matrix.mtx.gz")
In [5]: scipy.sparse.linalg.eigen(a.astype(np.float32))[0]
Out[5]:
array([ 1.00000119 +0.00000000e+00j, 0.99999952 +9.27230701e-07j,
0.99999952 -9.27230701e-07j, 1.00000107 +0.00000000e+00j,
1.00000715 +0.00000000e+00j, 1.00001323 +0.00000000e+00j])
In [6]: scipy.sparse.linalg.eigen(a.astype(np.float64))[0]
Out[6]:
array([ 0.+0.j, 0.+0.j, 0.+0.j, 0.+0.j, 0.+0.j, 0.+0.j])
Note that the 64-bit version takes about 10 times longer than the 32 bit version, and returns only zeros for the eigenvalues. This happens on NumPy 1.5.0, SciPy 0.8.0, Python 2.6.6 on 64-bit OS-X, where float64 is the default type of floats.
I think this is essentially a convergence issue. Calling the eigensolver with an extra argument tol=1e-4 will give reasonable results. While debugging this, I came up with a few questions that the docs don't quite answer, and it would be great if someone could help me out:
1. The sparse eigensolvers take a "maxiter" argument, which defaults to None. I would have expected this to mean that there is no cap on the amount of iterations. Looking at the code, it seems like it is set to 10 times the size of the matrix by default. Is this correct?
2. The eigensolver also takes an optional tolerance argument, which defaults to 0. This seems like an odd choice for an iterative numerical algorithm. The ARPACK manual mentions that the default tolerance is set to the machine precision, is this what a tolerance of 0 means? This might explain why I don't get reasonable results using 64 bit floats, because the precision is just too high.
3. Is there any way for the calling function to know whether convergence was achieved, or whether the eigensolver returned because the maxiter iterations were performed? Something like a "NotConverged" exception, maybe?
Hao, could you try to call the eigensolver with a generous tolerance argument, and see if this fixes your problem? If your issue is unrelated, I apologize for hijacking this thread.
Thanks,
Lutz
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