  Contour-integral-based rational filter leads to interior eigensolvers for
non-Hermitian generalized eigenvalue problems. Based on the Zolotarev's
problems, this paper proves the asymptotic optimality of the trapezoidal
quadrature of the contour integral in terms of the rational function
separation. A composite rule of the trapezoidal quadrature is derived. Two
interior eigensolvers are proposed based on the composite rule. Both
eigensolvers adopt direct factorization and multi-shift generalized minimal
residual method for the inner and outer rational functions, respectively. The
first eigensolver fixes the order of the outer rational function and applies
the subspace iteration to achieve convergence, whereas the second eigensolver
doubles the order of the outer rational function every iteration to achieve
convergence without subspace iteration. The efficiency and stability of
proposed eigensolvers are demonstrated on synthetic and practical sparse matrix
pencils.
