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Re: Eigenvalues of sparse arrays
*To*: mathgroup at smc.vnet.net
*Subject*: [mg102378] Re: Eigenvalues of sparse arrays
*From*: gopher <gophergoon at gmail.com>
*Date*: Fri, 7 Aug 2009 05:30:52 -0400 (EDT)
*References*: <200907310952.FAA19221@smc.vnet.net> <h50sjj$6pl$1@smc.vnet.net>
On Aug 3, 4:49 am, Mark McClure <mcmcc... at unca.edu> wrote:
> On Sun, Aug 2, 2009 at 5:59 AM, gopher <gopherg... at gmail.com> wrote:
> > Thanks for your reply. I read those docs but IMO they are somewhat
> > misleading. They state that the Arnoldi method may not converge but
> > not that it may converge and still give spurious eigenvalues.
>
> I agree; a warning should be issued. There are simple tests based on
> condition numbers for this sort of thing.
>
> The matrix I was using is here (it is only 156x156):
>
> >https://netfiles.uiuc.edu:443/aroy2/www/sparse%20array/bugmatrix.dat
>
> Arnoldi iteration is a technique to find the *largest* eigenvalue of a
> matrix. We can find other eigenvalues of the matrix A by applying Arnoldi
> iteration to the matrix (A-sI)^(-1), where s is a number close to the
> eigenvalue we want. This works since s+1/m is an eigenvalue of A close tos,
> whenever m is a large eigenvalue of (A-sI)^(-1). Not surprisingly, there is
> a problem if s is too close to an exact eigenvalue, for then A-sI is
> singular. A simple test for this possibility is the condition number of
> A-sI, which should not be too large. In your case, you are looking for
> eigenvalues close to zero of a matrix that is singular to start with. The
> condition number of this matrix is huge, since it's singular.
>
> dat = N[Import[
> "https://netfiles.uiuc.edu/aroy2/www/sparse%20array/bugmatrix.dat";]];
> sp = SparseArray[dat];
> LinearAlgebra`MatrixConditionNumber[sp]
> 7.22615*10^17
>
> We can fix the situation by using the Shift sub-option of the Arnoldi
> method. This options specifies the number s and should be set close to
> zero, but certainly not equal to zero. We can do this as follows.
>
> Eigenvalues[sp, 6,
> Method -> {"Arnoldi", "Shift" -> 0.01}]
> {0.382326, -0.382326, 0.350062, -0.350062, 4.74967*10^-15,
> 9.95731*10^-16}
>
> Eigenvalues[dat, -6]
> {0.382326, -0.382326, -0.350062, 0.350062, 2.29823*10^-15,
> 7.57673*10^-16}
>
> Hope that helps,
> Mark
Thanks very much, Mark and Roman. That was quite educational.
I am going to submit a bug report about this since it seems
Mathematica ought to give at least a warning when shift is too close
to an eigenvalue, given that the user cannot explicitly choose Lapack
over the Arnoldi method.
Abhishek
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