  Contour integration techniques have become a popular choice for solving the
linear and non-linear eigenvalue problems. They principally include the
Sakurai-Sugiura methods, the Beyn's algorithm, the FEAST/NLFEAST algorithms and
other rational filtering techniques. While these methods can result in
effective 'black-box' approach for solving linear eigenvalue problems, they
still present several shortcomings for addressing nonlinear eigenvalue problems
which are both mathematically and practically far more challenging. In this
paper, we introduce a new hybrid algorithm that advantageously combines the
iterative nature of NLFEAST with the effectiveness of Beyn's approach to deal
with general non-linearity. In doing so, this NLFEAST-Beyn hybrid algorithm can
overcome current limitations of both algorithms taken separately. After
presenting its derivation from both a Beyn's and NLFEAST's perspective, several
numerical examples are discussed to demonstrate the efficiency of the new
technique.
