finding eigenvalues of huge and very sparse matrix

I have the following problem. There is a matrix A of size NxN, where N = 200 000. It is very sparse, there are exactly M elements in each row, where M={6, 18, 40, 68, 102} (I have 5 different scenarios), the rest are zeros.

Now I would like to get all the eigenvalues and eigenvectors of matrix A.

Problem is, I cannot put matrix A into memory as it is around 160 GB of data. What I am looking for is a software that allows nice storing of sparse matrix (without zeros, my matrix is just few MB) and then putting this stored matrix without zeros to the algorithm that calculates eigenvalues and vectors.

Can any of you recommend me a software for that?

EDIT: I found out I can reconfigure my matrix A so it becomes a band matrix. Then I could use LAPACK to get the eigenvalues and eigenvectors (concretely: http://software.intel.com/sites/products/documentation/doclib/iss/2013/mkl/mklman/GUID-D3C929A9-8E33-4540-8854-AA8BE61BB08F.htm). Problem is, I need all the vectors, and since my matrix is NxN, I cannot allow LAPACK to store the solution (all eigenvectors) in the memory. The best way would be a function that will give me first K eigenvectors, then I rerun the program to get the next K eigenvectors and so on, so I can save the results in a file.

You may try to use the SLEPC library http://www.grycap.upv.es/slepc/description/summary.htm :

"SLEPc the Scalable Library for Eigenvalue Problem Computations, is a software library for the solution of large sparse eigenproblems on parallel computers."

Read the second chapter of their users'manual, "EPS: Eigenvalue Problem Solver". They are focused on methods that preserve sparcity...but a limited number of eigenvalues and eigenvectors are computed.

I hope our matrices have good properties (positive definite for instance...).

 EPSIsPositive(EPS eps,PetscBool *pos);

You may be interrested in "spectrum slicing" to compute all eigenvalues in a given interval... Or you may set a target and compute the closest eigenvalue around this target.

See http://www.grycap.upv.es/slepc/documentation/current/docs/manualpages/EPS/EPSSetWhichEigenpairs.html#EPSSetWhichEigenpairs

See examples http://www.grycap.upv.es/slepc/documentation/current/src/eps/examples/tutorials/index.html

Why do you need to compute all eigenvectors for such large matrices ?

Bye,