Graduate Efficiently Computing Eigenvalues of a Sparse Banded Matrix

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Efficient computation of eigenvalues for a sparse banded matrix, specifically a penta-diagonal Hamiltonian, is discussed. LAPACK routines, particularly dsyev and dsbev, are mentioned, though the latter did not yield significant improvements due to the distance of the outer bands. The limitations of LAPACK in leveraging sparsity for matrices with bands far from the diagonal are noted. ARPACK is suggested as an alternative, although its documentation is considered challenging to navigate. Exploring these options could enhance the efficiency of eigenvalue computations for the specified matrix structure.
Jimmy and Bimmy
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I have a Hamiltonian represented by a penta-diagonal matrix

The first bands are directly adjascent to the diagonals. The other two bands are N places above and below the diagonal.

Can anyone recommend an efficient algorithm or routine for finding the eigenvalues and eigenvectors?
 
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DrClaude said:
LAPACK can handle banded matrices. See for instance:
http://www.netlib.org/lapack/lug/node48.html
http://www.netlib.org/lapack/lug/node124.html
Thanks for the reply.

I have been using dsyev from the lapack routine (which is for symmetric matrices). I switched to dsbev (which is for symmetric band matrices), but didn't see much improvement. This is because, even though I only have 5 bands (diagonal + 2 upper + 2 lower), the outer 2 bands are a good distance away, and lapack doesn't take advantage of spare matrices.

Someone recommended arpack, but the documentation can be hard to follow.
 

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