The calculation of a segment of eigenvalues and their corresponding eigenvectors of a Hermitian matrix or matrix pencil has many applications. A new density-matrix-based algorithm has been proposed recently and a software package FEAST has been developed. The density-matrix approach allows FEAST's implementation to exploit a key strength of modern computer architectures, namely, multiple levels of parallelism. Consequently, the software package has been well received, especially in the electronic structure community. Nevertheless, theoretical analysis of FEAST has lagged. For instance, the FEAST algorithm has not been proven to converge. This paper offers a detailed numerical analysis of FEAST. In particular, we show that the FEAST algorithm can be understood as an accelerated subspace iteration algorithm in conjunction with the Rayleigh-Ritz procedure. The novelty of FEAST lies in its accelerator which is a rational matrix function that approximates the spectral projector onto the eigenspace in question. Analysis of the numerical nature of this approximate spectral projector and the resulting subspaces generated in the FEAST algorithm establishes the algorithm's convergence. This paper shows that FEAST is resilient against rounding errors and establishes properties that can be leveraged to enhance the algorithm's robustness. Finally, we propose an extension of FEAST to handle non-Hermitian problems and suggest some future research directions.
Non-Linear Least-Squares Optimization of Rational Filters for the Solution of Interior Eigenvalue Problems
RWTH Aachen University（亚琛工业大学）；Jülich Supercomputing Centre, Institute for Advanced Simulation, Forschungszentrum Jülich GmbH（于利希超级计算中心，先进模拟研究所，于利希研究中心有限公司）
Block algorithms with augmented Rayleigh-Ritz projections for large-scale eigenpair computation
Beijing International Center for Mathematical Research, Peking University（北京大学数学科学中心）；Department of Computational and Applied Mathematics, Rice University（莱斯大学计算与应用数学系）
