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The calculation of a segment of eigenvalues and their correspondingeigenvectors of a Hermitian matrix or matrix pencil has many applications. Anew density-matrix-based algorithm has been proposed recently and a softwarepackage FEAST has been developed. The density-matrix approach allows FEAST'simplementation to exploit a key strength of modern computer architectures,namely, multiple levels of parallelism. Consequently, the software package hasbeen well received, especially in the electronic structure community.Nevertheless, theoretical analysis of FEAST has lagged. For instance, the FEASTalgorithm has not been proven to converge. This paper offers a detailednumerical analysis of FEAST. In particular, we show that the FEAST algorithmcan be understood as an accelerated subspace iteration algorithm in conjunctionwith the Rayleigh-Ritz procedure. The novelty of FEAST lies in its acceleratorwhich is a rational matrix function that approximates the spectral projectoronto the eigenspace in question. Analysis of the numerical nature of thisapproximate spectral projector and the resulting subspaces generated in theFEAST algorithm establishes the algorithm's convergence. This paper shows thatFEAST is resilient against rounding errors and establishes properties that canbe leveraged to enhance the algorithm's robustness. Finally, we propose anextension of FEAST to handle non-Hermitian problems and suggest some futureresearch directions.

FEAST as a Subspace Iteration Eigensolver Accelerated by Approximate Spectral Projection