We present a rational filter for computing all eigenvalues of a symmetric definite eigenvalue problem lying in an interval on the real axis. The linear systems arising from the filter embedded in the subspace iteration framework, are solved via a preconditioned Krylov method. The choice of the poles of the filter is based on two criteria. On the one hand, the filter should enhance the eigenvalues in the interval of interest, which suggests that the poles should be chosen close to or in the interval. On the other hand, the choice of poles has an important impact on the convergence speed of the iterative method. For the solution of problems arising from vibrations, the two criteria contradict each other, since fast convergence of the eigensolver requires poles to be in or close to the interval, whereas the iterative linear system solver becomes cheaper when the poles lie further away from the eigenvalues. In the paper, we propose a selection of poles inspired by the shifted Laplace preconditioner for the Helmholtz equation. We show numerical experiments from finite element models of vibrations. We compare the shifted Laplace rational filter with rational filters based on quadrature rules for contour integration.

翻译：本文提出了一种有理滤波器，用于计算实轴上某一区间内对称正定特征值问题的所有特征值。该滤波器嵌入子空间迭代框架后产生的线性系统，通过预条件Krylov方法求解。滤波器极点的选择基于两个准则：一方面，滤波器应增强目标区间内的特征值，这表明极点应选在区间附近或内部；另一方面，极点的选择对迭代方法的收敛速度有重要影响。对于振动问题，这两个准则相互矛盾：特征值求解器的快速收敛要求极点位于区间内或附近，而迭代线性系统求解器在极点远离特征值时计算成本更低。本文受亥姆霍兹方程的平移拉普拉斯预条件器启发，提出了一种极点选择策略。我们通过振动问题的有限元模型进行了数值实验，并将平移拉普拉斯有理滤波器与基于围道积分求积准则的有理滤波器进行了比较。