[SciPy-dev] Comments on API for Matlab's eigs equivalent (computing a few eigenvalues only)

[josef.pktd at gmail.com]

On Thu, Feb 4, 2010 at 8:27 PM, David Cournapeau <cournape at gmail.com> wrote:
> On Fri, Feb 5, 2010 at 9:43 AM, Warren Weckesser
> <warren.weckesser at enthought.com> wrote:
>
>> David C,
>>
>> You said this was for symmetric matrices, but do you envision later
>> allowing nonsymmetric matrices?
>
> Yes. I have only implemented symmetric/Hermitian because that's the
> only solver that handles getting only a few eigenvalues in LAPACK.
> Matlab own eigs function uses ARPACK for both symmetric/unsymmetric
> cases, which is not good according to one of the Lapack developer
> (http://mail.scipy.org/pipermail/scipy-dev/2007-March/006778.html).
> But we could use ARPACK for the general case in eigs.
>
>>
>> If not, then perhaps the name of the function should be 'eigsh',
>> following the precedent set by numpy.linalg and scipy.linalg.
>
> I wonder whether those eigh functions are a good idea: I fear that
> most people will always use eig - maybe one could use the underlying
> eig*h solver in eig* if the matrix is detected as being symmetric ? I
> am not really knowledgeable about those issues, though. For example, I
> don't know whether the symmetric aspect should be checked exactly, or
> if it is better to use a symmetric solver  even if there are very
> small errors in say A'-A.
>
>>
>> In particular, the choice of ordering by magnitude or by real part is
>> convenient.
>
> It seems that one would need to implement the non-symmetric
> capabilities to sort this out. I fear that those options are solver
> specific, though - maybe the solution is to have two levels of API,
> one low-level and one high level.
>
> cheers,
>
> David
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the current version of scipy.linalg.eigh sorts from smallest to largest

>>> scipy.linalg.eigh(x)[0]
array([  4.04457427e-03,   1.84073286e-01,   6.74875960e-01,
         3.23328824e+00,   4.00741304e+00,   6.98333680e+00,
         1.45314842e+01,   1.49260377e+01,   2.47166702e+01,
         2.98955755e+01])
>>> scipy.linalg.eigh(x, eigvals=(0,3))[0]
array([ 0.00404457,  0.18407329,  0.67487596,  3.23328824])
>>> scipy.linalg.eigh(x, eigvals=(len(x)-3, len(x)-1))[0]
array([ 14.92603769,  24.71667021,  29.89557547])


>>> x = np.random.randn(10, 10) + 1j*np.random.randn(10, 10)
>>> x = np.dot(x.T, x)
>>> scipy.linalg.eigh(x, eigvals=(0,3))[0]
array([-36.82039662, -18.20362967, -12.21716065,  -9.79752274])
>>> scipy.linalg.eigh(x, eigvals=(len(x)-3, len(x)-1))[0]
array([ 14.44815917,  28.8394026 ,  42.45471058])

Josef