FEAST as a Subspace Iteration Eigensolver Accelerated by Approximate Spectral Projection

TL;DR

This paper provides a detailed numerical analysis of the FEAST algorithm, demonstrating its convergence as an accelerated subspace iteration method and extending its applicability to non-Hermitian problems.

Contribution

It offers the first rigorous convergence proof for FEAST and introduces an extension for non-Hermitian eigenvalue problems.

Findings

FEAST is equivalent to an accelerated subspace iteration with Rayleigh-Ritz.

The spectral projector approximation ensures convergence.

FEAST is robust against rounding errors.

Abstract

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.