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.

翻译：基于围道积分的有理滤波方法可用于非厄米广义特征值问题的内部特征值求解。本文依据佐洛塔廖夫问题，证明了从有理函数分离角度而言，围道积分的梯形求积具有渐近最优性。进一步推导出梯形求积的复合准则，并基于该准则提出了两种内部特征值求解器。两种求解器分别对内层和外层有理函数采用直接分解与多移位广义最小残差法。第一种求解器固定外层有理函数的阶数，通过子空间迭代实现收敛；第二种求解器则在每次迭代中加倍外层有理函数的阶数，无需子空间迭代即可收敛。通过合成及实际稀疏矩阵束的数值实验，验证了所提求解器的效率与稳定性。