Projective block Lanczos algorithm for dense, Hermitian eigensystems (Q1913745)

The authors introduce the projective block Lanczos (PBL) algorithm for seeking extreme eigenvalues and eigenvectors of an \(N\times N\) Hermitian matrix and present examples and applications on various physical problems. The first example is drown from many-body quantum mechanics, specifically, clusters of magnetic dipoles. This one leads to sparse Hermitian matrices. Results are given for \(N\leq 16384\). The second example is the density functional theory of diatomic beryllium and \(\text{Be}_{13}\) clusters, using plane waves and including all electrons. This example is a dense, complex Hermitian, with all matrix elements nonzero. The goal of the research in this area is a block Lanczos algorithm using an accurate shift operator for large, dense eigensystem implemented on a distributed memory multiprocessor. The (here published) PBL algorithm and the evaluation of approximate spectral transforms are a step towards this goal for solving specific classes of physical problems.