We present a greedy algorithm for computing selected eigenpairs of a large sparse matrix H that can exploit localization features of the eigenvector. When the eigenvector to be computed is localized, meaning only a small number of its components have large magnitudes, the proposed algorithm identifies the location of these components in a greedy manner, and obtains approximations to the desired eigenpairs of H by computing eigenpairs of a submatrix extracted from the corresponding rows and columns of H. Even when the eigenvector is not completely localized, the approximate eigenvectors obtained by the greedy algorithm can be used as good starting guesses to accelerate the convergence of an iterative eigensolver applied to H. We discuss a few possibilities for selecting important rows and columns of H and techniques for constructing good initial guesses for an iterative eigensolver using the approximate eigenvectors returned from the greedy algorithm. We demonstrate the effectiveness of this approach with examples from nuclear quantum many‐body calculations, many‐body localization studies of quantum spin chains and road network analysis.

中文翻译：

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带有局部特征向量的对称矩阵特征值的贪心算法

我们提出了一种贪婪算法，用于计算大型稀疏矩阵H的选定特征对，该特征对可以利用特征向量的定位特征。当将要计算的特征向量定位时，意味着只有少量分量具有较大的幅度，所提出的算法以贪婪的方式识别这些分量的位置，并通过计算子矩阵的特征对获得与H的期望特征对的近似值。从H的相应行和列中提取。即使在特征向量未完全定位的情况下，通过贪婪算法获得的近似特征向量也可以用作良好的开始猜测，以加速应用于H的迭代特征求解器的收敛。我们讨论了选择H的重要行和列的几种可能性，以及使用贪婪算法返回的近似特征向量为迭代特征求解器构造良好的初始猜测的技术。我们通过核量子多体计算，量子自旋链的多体本地化研究以及路网分析等示例来证明这种方法的有效性。