Re: [eigen] ordering of eigenvalues of EigenSolver

[Susanne Suter <susanne.suter@xxxxxxxxx>]

Hi

Thanks Robert for pointing out my question, which was about sorting
according to absolute value or signed value. Because in the current
implementation the eigenvalues are ordered taking into account the
sign (my example: -2.77687, -1.50582, -0.0370092,  0.848378 (last two
values should be switched or if ascending order, the whole ordering
should be reversed).

I'm actually using LAPACK SVD now, but I was considering to use eigen
library for SVD and/or Eigenvalue decomposition. So, I'm verifying if
eigen produces the results I expect.

I think we should draw a differentiation between SVD and eigenvalue
decomposition:

For SVD, the solution is unique except for the sign and the singular
values are always ordered descending. I.e., the singular values are
ordered according to their absolute value.

To my understanding, for eigenvalue decomposition the values don't
necessarily have to be ordered. But I can only think of applications,
which use them ordered and when ordering them, I would use it
according to absolute value (for real value problems). I.e., use the x
largest magnitude values.

Furthermore, other eigenvalue decomposition libraries seem to order
eigenvalues, too. E.g., MatLab (based on ARPACK) orders the eigenvalue
according to absolute value and it seems that LAPACK SSYEV does the
same but in reverse order.

So, is there a bug in the eigen library or are the values intended to
be sorted like that (according to weight and direction/sign)?

Thank you and best,
Susanne

On Mon, Feb 21, 2011 at 10:20 AM, Robert Bocquier
<robert.bocquier@xxxxxxxxxxx> wrote:
> Hi Benoit,
>
> I am interested in your answer to Susanne question, but I think you
> didn't fully address it.
> Her question wasn't about ascending versus descending ordering, but was
> about the basis for the sort. Is it (and should it be) the eingenvalues
> directly, or the absolute values of the eigenvalues ?
>
> Thx
> Robert
>
> Le 18/02/2011 16:59, Benoit Jacob a écrit :
>> Hi,
>>
>> Sorry for the belated answer, I hadn't actually checked carefully what
>> both Eigen and others (LAPACK) were doing until today.
>>
>> As it turns out, we are doing exactly the same thing as LAPACK:
>>
>>  * for self-adjoint eigensolver, we sort eigenvalues in ascending
>> order. For LAPACK, see:
>>     http://www.netlib.org/lapack/single/ssyev.f
>>
>>  * for SVD, we sort singular values in descending order. For SVD, see:
>>     http://www.netlib.org/lapack/single/sgesvd.f
>>
>> I agree that it's pretty weird to be using sometimes ascending and
>> sometimes descending order. But since that's what both LAPACK and
>> ourselves have been doing, we shouldn't change that now.
>>
>> Benoit
>>
>> 2011/2/8 Susanne Suter <susanne.suter@xxxxxxxxx>:
>>> Hi
>>>
>>> Sorry, I accidently hit the "send button" too early. Here my complete
>>> messge again.
>>>
>>> I'm testing the Eigen library in order to use it for eigenvalue
>>> decomposition or SVD. In the end I need the eigenvectors (left
>>> singular vectors) and the eigenvalues (singular values).
>>>
>>> I noticed that when using the EigenSolver classes, the ordering of the
>>> eigenvalues is not as I expected. Normally, I would expect that I get
>>> an ordering analogous to the singular values, i.e., an ordering with
>>> the maximum absolute value first and then decreasing values ordered by
>>> their absolute value (sometimes also called "largest magnitude
>>> eigenvalues"). What I get now, looks a bit like an ordering
>>> considering the sign (SelfAdjointEigenSolver). I had similar issues
>>> with EigenSolver and ComplexEigenSolver (however, I found the ordering
>>> not consistent in all methods).
>>>
>>> ### SelfAdjointEigenSolver: eigenvalues of A are:    -2.77687
>>> -1.50582 -0.0370092   0.848378
>>> ### ComplexEigenSolver: eigenvalues are: (-0.0370093,0) (0.848378,0)
>>> (-1.50582,0) (-2.77687,0)
>>> ### EigenSolver:  eigenvalues are : (-2.77687,0) (-0.0370092,0)
>>> (0.848379,0)   (-1.50582,0)
>>>
>>> However, I would expect
>>> ### eigenvalues of A are:   -2.77687   -1.50582 0.848378 -0.0370092
>>>
>>> Until now, I only was using eigenvalue decompositions, which used the
>>> same ordering for eigenvalues as the SVD for the singular values (e.g.
>>> Matlab). Is there any reason why the implementation of Eigen is
>>> different? Or is there any option to change the ordering?
>>>
>>> I'm using the following test matrix:
>>>
>>> A =
>>>
>>>   -2.0000   -0.6714    0.8698    0.5792
>>>   -0.6714   -1.1242   -0.0365   -0.5731
>>>    0.8698   -0.0365   -0.4660   -0.8542
>>>    0.5792   -0.5731   -0.8542    0.1188
>>>
>>> and I test it with the following code (using the Eigen3 version).
>>>
>>>        Eigen::SelfAdjointEigenSolver<Eigen::Matrix4f> es;
>>>        es.compute(A);
>>>        std::cout << "### SELFADJOINT EIGENVALUE DECOMPOSITION ### " << std::endl;
>>>        std::cout << "** eigenvalues of A are: " <<
>>> es.eigenvalues().transpose() << std::endl;
>>>        std::cout << "** eigenvectors of A are: " << std::endl <<
>>> es.eigenvectors() << std::endl;
>>>
>>> Thank you for any hints or advice on that topic.
>>>
>>> Best,
>>> Susanne
>>>
>>>
>>>
>>
>>
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