  Calculating portions of eigenvalues and eigenvectors of matrices or matrix
pencils has many applications. An approach to this calculation for Hermitian
problems based on a density matrix has been proposed in 2009 and a software
package called 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 and subsequently commercialized. A
detailed theoretical analysis of Hermitian FEAST has also been established very
recently. This paper generalizes the FEAST algorithm and theory, for the first
time, to tackle non-Hermitian problems. Fundamentally, the new algorithm is
basic subspace iteration or Bauer bi-iteration, except applied with a novel
accelerator based on Cauchy integrals. The resulting algorithm retains the
multi-level parallelism of Hermitian FEAST, making it a valuable new tool for
large-scale computational science and engineering problems on leading-edge
computing platforms.
