  The FEAST algorithm, due to Polizzi, is a typical contour-integral based
eigensolver for computing the eigenvalues, along with their eigenvectors,
inside a given region in the complex plane. It was formulated under the
circumstance that the considered eigenproblem is Hermitian. The FEAST algorithm
is stable and accurate, and has attracted much attention in recent years.
However, it was observed that the FEAST algorithm may fail to find the target
eigenpairs when applying it to the non-Hermitian problems. Efforts have been
made to adapt the FEAST algorithm to non-Hermitian cases. In this work, we
develop a new non-Hermitian scheme for the FEAST algorithm. The mathematical
framework will be established, and the convergence analysis of our new method
will be studied. Numerical experiments are reported to demonstrate the
effectiveness of our method and to validate the convergence properties.
