Re: eigs compatibility

[Daniel J Sebald]

On 07/04/2013 02:21 PM, John W. Eaton wrote:
> On 07/04/2013 03:03 PM, Daniel J Sebald wrote:
>
> > I guess a sparse matrix of 2x2 isn't really all that sparse. Anyway, if
> > it is important to have compatibility or just handle all dimensions, and
> > it is only 2x2 and 1x1 that must be dealt with, rather than require a
> > different math package be used to offer those results, it might be just
> > as easy to write out the formulas for the solution to the eigenvalues.
> > If one were to study the math, it probably breaks down into some simple
> > inversion formulas which aren't so challenging until getting above 3x3.
> > Why isn't there support in the sparse math package?
>
> I have no clue why ARPACK has the limitation of providing at most N-2
> eigenvalues, but that's the way it works. I'm not going to try to fix
> that problem.

I'm not familiar with the ARPACK library, but it looks like the number 
of subdimensions is selectable as 'k'.  I'm going to guess that these 
eigenvalue techniques of ARPACK are fast at finding the eigenvalues of 
higher strength but then become slower with weaker eigenvalues.  So, if 
one were to ask for all the eigenvalues, the routine my get bogged down, 
in a sense.  Not sure.

But here

http://www.caam.rice.edu/software/ARPACK/UG/node13.html#SECTION00570000000000000000

the developers indicate that BLAS and LAPACK are requirements and if 
there is something in those libraries to substitute or utilize, do so. 
Given that, I guess falling back on those libraries for small matrix 
eigenvalues is fine.


> Apparently Matlab's eigs function automatically handles the cases of
> computing N or N-1 eigenvalues. I don't know how to do that other than
> by calling eig and sorting the results as needed.

Oh, so it isn't just the 2x2 case.  It might be if someone has a 20x20 
matrix and asks for 19 eigenvalues/vectors?  I would say try a timing 
experiment comparing:

20x20 non-sparse
20x20 sparse requesting 18 eigenvalues/vectors

If those two are close then, i.e., the ARPACK really slows down as the 
number of subdimensions approaches the dimension of the matrix, then the 
ARPACK developers probably figured it is best to have the users of the 
software fall back on the LAPACK or BLAS full matrix equivalent...as you 
are proposing.  I've a feeling that there really isn't a problem here, 
it just that the original Octave programmer didn't realize ARPACK 
suggests falling back on LAPACK or BLAS.


> I think Matlab does not issue a warning until N >= 8, then it says that
> you should use "eig (full (A))" or something similar, but still goes
> ahead and does the calculation anyway. Would someone like to confirm
> that behavior?
>
> I don't think it makes sense to try to write out the analyical solution
> for eigenvalues for N == 6, or 5, ...

One wouldn't want to write out analytical solutions for N equal 5 or 6, 
just 2 or in a pinch 3.  Anything above that is best handled with 
rotation, pivoting, whatever.

Also, I don't know why Matlab would issue a warning about N > 7.  That 
doesn't seem like a such a big and CPU consuming problem.  I would guess 
somewhere above 20 is when things become noticeably slow for modern 
processors.  Maybe 15, but certainly not 7.

Dan