Parallel eigenvalue calculation based on multiple shift-invert Lanczos and contour integral based spectral projection method

We discuss the possibility of using multiple shift-invert Lanczos and contour integral based spectral projection method to compute a relatively large number of eigenvalues of a large sparse and symmetric matrix. The key to achieving high parallel efficiency in this type of computation is to divide the spectrum into several intervals in a way that leads to optimal use of computational resources. We discuss strategies for dividing the spectrum. Our strategies make use of a eigenvalue distribution profile that can be estimated through inertia counts and cubic spline fitting. We also make use of a simple cost model that describes the cost of computing k eigenvalues within a single interval in terms of the asymptotic cost of sparse matrix factorization and triangular substitutions. Several computational experiments are performed to demonstrate the effect of spectrum division on the overall performance of both multiple shift-invert Lanczos and the contour integral based method. We also show parallel scalability of both approaches in the strong and weak sense. Our experiments indicate that multiple shift-invert Lanczos is generally more efficient than the contour integral based spectral projection method when implemented properly.