Interior eigensolver for sparse Hermitian definite matrices based on Zolotarev’s functions
Interior eigensolver for sparse Hermitian definite matrices based on Zolotarev’s functions
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基于 Zolotarev 函数的稀疏 Hermitian 定矩阵的内部特征求解器
DOI：
10.4310/cms.2021.v19.n4.a11
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发表时间：
2017
影响因子：
1
通讯作者：
Haizhao Yang
中科院分区：
文献类型：
--
作者：
Yingzhou Li;Haizhao Yang
本文提出了一种计算稀疏Hermitian定矩阵束$（A，B）$的选定广义特征对的有效方法。基于Zolotarev的最佳有理函数逼近的正负号函数和保角映射技术，我们构造的最佳有理函数逼近的矩形函数支持在任意区间上通过功能组合与部分分数表示。这种新的最佳有理函数逼近可以用来构造$（A，B）$的谱滤波器，其极点数目比直接构造的滤波器少。结合快速直接求解器和平移不变的广义最小残差法，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，所提出的有理函数逼近在特征值求解器中需要稀疏矩阵分解来求解多移位线性系统时更有效，因为在我们的方法中需要较少的矩阵分解。计算化学数值算例表明了该方法的有效性和稳定性。
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 the 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.