This paper proposes an efficient method for computing selected generalized eigenpairs of a sparse Hermitian definite matrix pencil $(A,B)$. Based on Zolotarev’s best rational function approximations of the signum function and conformal mapping techniques, we construct the best rational function approximation of a rectangular function supported on an arbitrary interval via function compositions with partial fraction representations. This new best rational function approximation can be applied to construct spectrum filters of $(A,B)$ with a smaller number of poles than a direct construction without function compositions. Combining fast direct solvers and the shift-invariant generalized minimal residual method, a hybrid fast algorithm is proposed to apply spectral filters efficiently. Compared to the state-of-the-art algorithm FEAST, the proposed rational function approximation is more efficient when sparse matrix factorizations are required to solve multi-shift linear systems in the eigensolver, since a smaller number of matrix factorizations is needed in our method. The efficiency and stability of the proposed method are demonstrated by numerical examples from computational chemistry.

中文翻译：

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基于 Zolotarev 函数的稀疏 Hermitian 定矩阵的内部特征求解器

本文提出了一种计算稀疏 Hermitian 定矩阵 pen $(A,B)$ 的选定广义特征对的有效方法。基于 Zolotarev 的符号函数的最佳有理函数近似和保形映射技术，我们通过具有部分分数表示的函数组合构造了支持任意区间的矩形函数的最佳有理函数近似。这种新的最佳有理函数逼近可用于构造 $(A,B)$ 的频谱滤波器，与没有函数组合的直接构造相比，极点数更少。结合快速直接求解器和平移不变广义最小残差方法，提出了一种混合快速算法来有效地应用谱滤波器。与最先进的算法 FEAST 相比，当需要稀疏矩阵分解来解决特征求解器中的多移位线性系统时，所提出的有理函数近似更有效，因为我们的方法需要较少数量的矩阵分解。计算化学的数值例子证明了所提出方法的效率和稳定性。