A parallel algorithm for computing partial spectral factorizations of matrix pencils via Chebyshev approximation

Abstract

We propose a distributed-memory parallel algorithm for computing all 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 \texttt{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.