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my question is similar to

how to diagonalize a large sparse symmetric matrix, to get the eigenvalues and eigenvectors

however i wish to be more concrete and ask if one can, on a standard PC (e.g. a 500$ laptop :)) and using existing libraries, find all eigenvalues of a 50,000x50,000 sparse (up to 5 entries in a row) symmetric positive semidefinite matrix (to be again more concrete, a graph Laplacian), and how reliable this computation might be.

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Are you familiar with the usual methods? First reduce the matrix to tridiagonal form using orthogonal similarity (possibly Givens rotations will be more efficient in your application than Householder transformations). Then apply shifted-QR or Lanczos methods to isolate the eigenvalues. Since these are iterative methods, and you can always return to your original matrix for refinement, I don't see a reliability issue. However you are asking for 50,000 eigensolutions, and that's going to be a long running calculation. –  hardmath Feb 28 '13 at 13:59
    
thank you, I cannot upvote your comment for technical reasons, but for me it is a helpful information. –  Giovanni Rossi Feb 28 '13 at 14:04
    
Is this a 2D Laplacian on a regular grid? –  hardmath Feb 28 '13 at 14:05
    
For Questions in Math.SE it's better to focus on the math aspects of your problem rather than hardware or software issues; those will likely be considered off-topic in this forum. –  hardmath Feb 28 '13 at 14:10
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There is a serious problem with the "usual methods", in that they do not make use of the fact that your matrices are sparse. With minimal googling I found a survey at grycap.upv.es/slepc/documentation/reports/str6.pdf. –  Chris Godsil Feb 28 '13 at 14:13

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