A Parallel Algorithm for Computing Partial Spectral Factorizations of Matrix Pencils via Chebyshev Approximation

.We propose a distributed-memory parallel algorithm for computing some of the algebraically smallest eigenvalues (and corresponding eigenvectors) of a large, sparse, real symmetric positive definite matrix pencil that lie within a target interval. The algorithm is based on Chebyshev interpolation of the eigenvalues of the Schur complement (over the interface variables) of a domain decomposition reordering of the pencil and accordingly exposes two dimensions of parallelism: one derived from the reordering and one from the independence of the interpolation nodes. The new method demonstrates excellent parallel scalability, comparing favorably with PARPACK, and does not require factorization of the mass matrix, which significantly reduces memory consumption, especially for 3D problems. Our implementation is publicly available on GitHub.Keywordssymmetric generalized eigenvalue problemspectral Schur complementsChebyshev approximationparallel computingMSC codes15A1865D1565F1565N5565Y0568W10