Special method for finding eigenvalues of a large sparse symmetric matrix?

10% fill is huge. There is usually no point in using a sparse solver if you have that many nonzero entries. (On the order of 10 nonzero entries per row is common in real sparse-matrix applications, which corresponds to < 0.1% fill for a 14000x14000 matrix.)

And sparse-direct solvers will generally perform poorly for random fill. The algorithms work best for sparsity that comes from local interactions on a mesh or grid.

1d grids are best, and 2d grids/meshes are still very good. For 3d grids, the scaling of sparse-direct methods is worse (this is a consequence of the difficulty of partitioning meshes in higher dimensions, essentially…see e.g. Davis’ book on sparse-direct methods). For very large 3d problems you usually have to switch to purely iterative methods (GMRES, CG, etc) and hope you can find a good preconditioner.

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