According to this link, All eigenpairs of large sparse symmetric matrix. The guy @Baranas seems to have given a very confident answer about solving the whole Eigen spectrum. May I know if anyone has any suggestions on what he is talking about or suggestions on what software to use (currently thinking about SLEPc) that allows computation to be distributed across a few nodes on the supercomputer? My sparse matrix is only around 2% non-null, and ideally, all eigenvalues/spectrums could be found, but if not the smallest 10% eigenvalues. Have tried the Python scipy.sparse.linalg.eigsh, but it gives great speed for large eigenvalues, for small eigenvalues spectral shift results in some delay, and the time is comparable to dense matrix calculation.
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Storing all eigenvectors requires $O(N^2)$ space, which is not feasible for large-scale problems. Of course, neither is computing them since that has a complexity that must at least be as bad. – Wolfgang Bangerth Aug 28 '23 at 22:18