Re: Alternatives to ARPACK?
- To: mathgroup at smc.vnet.net
- Subject: [mg108353] Re: Alternatives to ARPACK?
- From: Jim Lambaugh <lambaugh at gmail.com>
- Date: Sun, 14 Mar 2010 06:01:00 -0500 (EST)
- References: <hmo27i$qap$1@smc.vnet.net>
On 4 Mar., 11:32, Matin Inck <matin.i... at gmail.com> wrote:
> Hi guys
>
> My system is currently described by a dense 2500x2500 matrix, which
> contains many zero entries. Diagonalizing this matrix using
> Eigensystem takes about 2 minutes, which seems like a long time, and I
> assume it has got something to do with the fact that the eigenvalues
> are degenerate. In order to investigate my system further, I need to
> generate a larger matrix (say 3600x3600), but my computer runs out of
> memory when I try to diagonalize it in this case (Mathematica
> complaints about no memory).
>
> I have two questions.
>
> 1) If I want the full spectrum of the matrix, what options do I have?
>
> 2) It might be possible that I can do with only a small part of the
> spectrum of my matrix. In this case what I would use is following
> piece of code, where =E2=80=9CHamiltonian=E2=80=9D is the name of the matrix:
>
> Eigensystem[SparseArray[Hamiltonian], 50, Method=EF=82=AE{"Arnoldi",
> "Shift"=EF=82=AE0.25}]
>
> Just to be absolutely sure: Does this give me the 50 eigenvalues (and
> corresponding eigenvectors) closest to 0.25? Also, I believe I read
> somewhere that for each time an eigenvector/value is found using this
> method, there is a loss of precision. Can you confirm this?
>
> Any help would be much appreciated.
>
> Best,
> Matin.
Yes, that code finds the eigenvalues closest to 0.25. Regarding the
precision, I can't be of much help, but I also need to find out what
precision Mathematica finds eigenvalues with when using Arnoldi-
methods.What I found in the user guide is "The numerical accuracy of
the computed eigenvalues and vectors is user specified", but I can't
find any options in Mathematica specifying this. I will let you know
if I find out more.
Hope this helps.