Interior Eigensolver Based on Rational Filter with Composite rule

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.