Re: svds test failure

[Ed Meyer]

On Tue, Jul 31, 2012 at 1:41 PM, Daniel J Sebald <address@hidden> wrote:

On 07/31/2012 03:23 PM, Ed Meyer wrote:

On Tue, Jul 31, 2012 at 12:31 PM, Daniel J Sebald

<address@hidden <mailto:address@hidden>> wrote:

    On 07/31/2012 02:22 PM, Ed Meyer wrote:

        As I pointed out in the "make check" thread the problem is that we
        should not be
        using absolute tolerances because that does not account for the
        size of
        the matrix
        elements.  What we should do is to estimate if the result lies
        in the
        interval we would
        get with roundoff perturbations in the matrix.  If it does,
        that's as
        good as can be expected.
        Eispack used to do something like that with their "performance
        index"
        but I think you can
        get better estimates.


    Give us an idea of what code you are suggesting Ed.  Something like

    eps*size(s,1)*size(s,2)

    ?  Something concerning the rank of the input matrix?

    Dan

    PS: Did you mean to CC to the maintainers list?


oops - I did mean to CC
I'm attaching a page from a manual I wrote describing how I judge
results.  It's more
involved than the old eispack tests and maybe more work than we want to do.
Much simpler would be to just use the norm of the matrix times some
factor times
machine epsilon instead of the absolute tolerances.

We're talking the linear case here, so what does equation (7) become in that case?  A'(lambda) should be simply a constant matrix?  Does that then turn out to be Frobenius norm or something?  So

for the linear case A(lambda) = B - lambda*I where B is the matrix who's svd we want
 

tol = 100*eps*norm(A, 'fro');

and

%!testif HAVE_ARPACK
%! s = svds (speye (10));

%! assert (s, ones (6, 1), tol);

Something like that?
 

And what of the degenerative case of

%!testif HAVE_ARPACK
%! [u2,s2,v2,flag] = svds (zeros (10), k);
%! assert (u2, eye (10, k));
%! assert (s2, zeros (k));
%! assert (v2, eye (10, 7));

where the norm is zero?  Just leave as is, because we aren't really testing the performance in this case so much as the library's ability to handle the degenerate case?

Dan

 

to handle the case of a matrix with zero (or small) elements we could use

  tol = 100*eps*min (1.0, norm(A,"fro"))

The results from ARPACK should not depend on the starting vector; if they do there is probably
something wrong with the installation. Which brings up another point - computing an svd by solving
a (larger) eigenvalue problem is probably not the best way.  There are sparse svd solvers out
there that might be more efficient and robust.  I'll try a few.

As for test cases I put together a case to create randomly sized and shaped matrices which
are very ill-conditioned to test the algorithm;  I'll use it to compare svd solvers.  I was surprised
that Arnoldi did not converge for some test cases.
-- 
Ed Meyer