Re: [R] highest eigenvalues of a matrix

From: baptiste Auguié <ba208_at_exeter.ac.uk>
Date: Thu, 19 Jun 2008 20:49:02 +0100

Thank you all for your interest in this. The ARPACK library looks like a very interesting option, I wish I had some experience in interfacing R with such libraries. I'll try to learn on a simple example first, and if successful I will try to adapt the igraphlib interface to provide the desired options ( for the record, the original problem was to find the first 3 eigenvalues (decreasing magnitude) and corresponding eigenvectors of a 200x200 Hermitian matrix by an iterative process). Maybe a more general approach would be better, but I'm already definitely out of my league here (not to mention the incompatibility issue with LAPACK already mentioned).

Many thanks,

baptiste

On 19 Jun 2008, at 19:08, Katharine Mullen wrote:

> On Thu, 19 Jun 2008, Martin Maechler wrote:
>
>>>>>>> "KateM" == Katharine Mullen <kate_at_few.vu.nl>
>>>>>>> on Thu, 19 Jun 2008 11:50:22 +0200 (CEST) writes:
>>
>> KateM> On Thu, 19 Jun 2008, Simon Wood wrote:
>>>>>
>>>>> I happily use eigen() to compute the eigenvalues and
>>>> eigenvectors of > a fairly large matrix (200x200, say),
>>>> but it seems over-killed as its > rank is limited to
>>>> typically 2 or 3. I sort of remember being taught > that
>>>> numerical techniques can find iteratively decreasing
>>>> eigenvalues > and corresponding orthogonal eigenvectors,
>>>> which would provide a nice > alternative (once I have the
>>>> first 3, say, I stop the search).
>>>>
>>>> Lanczos iteration will do this efficiently (see
>>>> e.g. Golub & van Loan "Matrix Computations"), but I don't
>>>> think that there are any such routines built into R or
>>>> LAPACK (although I haven't checked the latest LAPACK
>>>> release). When I looked it seemed that the LAPACK options
>>>> that allow you to select eigen-values/vectors still
>>>> depend on an initial O(n^3) decomposition of the matrix,
>>>> rather than the O(n^2) that a Lanczos based method would
>>>> require.
>>>>
>>
>> KateM> ARPACK (http://www.caam.rice.edu/software/ARPACK/) uses a
>> KateM> Lanczos method for symmetric matrics; otherwise it
>> uses an
>> KateM> Arnoldi iteration. Development of an R interface to
>> ARPACK
>> KateM> would be a nice project (but unfortunately one I don't
>> have
>> KateM> time for for a while). Maybe one of the maintainers of a
>> KateM> package for sparse matrices would be interested.
>>
>> Yes, thank you, Kate, we have been.
>> The nice thing in ARPACK is its concept of "reverse
>> communication", such that we should be able to use it both
>> for sparse and for dense matrices.
>> We would only have to provide a "function" for computing
>> "A %*% x" and hence could use different ones, depending on the
>> class of A.... all in theory at least.
>>
>> One problem I see with the with that code is that its README file
>> contains
>>
>> *** NOTE *** Unless the LAPACK library on your system is
>> version 2.0,
>> we strongly recommend that you install the LAPACK routines
>> provided with
>> ARPACK. Note that the current LAPACK release is version 3.0; if
>> you are
>> not sure which version of LAPACK is installed, pleaase compile
>> and link
>> to the subset of LAPACK included with ARPACK.
>>
>> i.e. you need to use old/outdated Lapack with ARPACK instead of
>> the current Lapack . Something that sheds a bad light to ARPACK
>> in my view.
>>
>
> Dear Martin,
>
> I agree this is not ideal. I wonder why it has not been kept up to
> date
> with LAPACK and how broken it is with version 3.1.1. (but can't
> experiment
> now).
>
>> In the mean time (I've suffered from a computer crash and other
>> interruptions) I see Gabor has already mentioned the following
>>
>> Then, we have seen that the 'igraph' package als uses a port of
>> (part of) ARPACK as part of the C++ 'igraphlib' library.
>>
>>
>>
>>>> My `mgcv' package (see cran) uses Lanczos iteration for
>>>> setting up low rank bases for smoothing. The source code
>>>> is in mgcv/src/matrix.c:lanczos_spd, but I'm afraid that
>>>> there is no direct R interface, although it would not be
>>>> too hard to write a suitable wrapper. It requires the
>>>> matrix to be symmetric.
>>>>
>>>>> Looking at the R source code for eigen and some posts
>>>> on this list, > it seems that the function uses a LAPACK
>>>> routine, but obviously all > the options are not
>>>> available through the R wrapper. I'm not > experienced
>>>> enough to try and make my own interface with Fortran >
>>>> code, so here are two questions:
>>>>>
>>>>> - is this option (choosing a desired number of
>>>> eigenvectors) already > implemented in some function /
>>>> package that I missed? --- In the symmetric case you can
>>>> use `svd' which lets you select (although you'd need to
>>>> fix up the signs of the singular values to get
>>>> eigen-values if the matrix is not +ve definite). But this
>>>> answer is pretty useless as it will be slower than using
>>>> `eigen' and getting the full decomposition.
>>>>
>>>> Of course if you know that your matrix is low rank
>>>> because it's a product of non-square matrices then
>>>> there's usually some way of getting at the
>>>> eigen-decomposition efficiently. E.g. if A=B'B where B is
>>>> 3 by 1000, then the cost can easily be kept down to
>>>> O(1000^2) in R...
>>>>
>>>> best, Simon
>>>>
>>>>> - is the "range of indices" option in DSYEVR.f <
>>>> http:// > www.netlib.org/lapack/double/dsyevr.f > what I
>>>> think, the indices of > the desired eigenvalues ordered
>>>> from the highest to lowest?
>>>>>
>>
>> KateM> dsyevr.f will work with symmetric real matrices only.
>> When the range
>> KateM> argument of dysevr is set to 'I', arguments il and iu
>> KateM> seem to specify the range of eigenvalue indices you
>> KateM> want in ascending order (lowest to highest, not
>> KateM> highest to lowest). If you look at
>> KateM> https://svn.r-project.org/R/trunk/src/modules/lapack/
>> Lapack.c
>> KateM> you see that range is always set to 'A'.
>>
>> yes, but do you agree that using 'I' (and a range) does not
>> really help much, since the factorization used is the "full" one
>> anyway ?
>>
>
> Now I see -- I had read
>
> Eigenvalues and eigenvectors can be
> * selected by specifying either a range of values or a range of
> * indices for the desired eigenvalues.
>
> with _selected_ magically replaced by _computed_. Thanks.
>
>>
>>>>> Many thanks in advance for any piece of advice,
>>
>>>>> Sincerely,
>>>>>
>>>>> Baptiste
>>>>>
>>>>> dummy example if needed:
>>>>>
>>>>> test <- matrix(c(1, 2, 0, 4, 5, 6, 1.00001, 2, 0),
>>>> ncol=3) > eigen(test)
>>>>>
>>>>>
>>>>>
>>>>>
>>>>> _____________________________
>>>>>
>>>>> Baptiste Auguié
>>>>>
>>>>> Physics Department > University of Exeter > Stocker
>>>> Road, > Exeter, Devon, > EX4 4QL, UK
>>>>>
>>>>> Phone: +44 1392 264187
>>>>>
>>>>> http://newton.ex.ac.uk/research/emag >
>>>> http://projects.ex.ac.uk/atto
>> [............]
>>
>>
>>>> --
>>>>> Simon Wood, Mathematical Sciences, University of Bath,
>>>> Bath, BA2 7AY UK > +44 1225 386603
>>>> www.maths.bath.ac.uk/~sw283
>>
>
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Baptiste Auguié

Physics Department
University of Exeter
Stocker Road,
Exeter, Devon,
EX4 4QL, UK

Phone: +44 1392 264187

http://newton.ex.ac.uk/research/emag
http://projects.ex.ac.uk/atto



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