标题： 预处理迭代方法用于特征值计数 显示英文标题

标题： Preconditioned iterative methods for eigenvalue counts

摘要： 我们描述了预处理迭代方法，用于估计 Hermite 矩阵在给定区间内的特征值个数。 这种估计在许多应用中都很有用。特别是，它可以用来开发一种有效的谱切片策略，以计算 Hermite 矩阵的许多特征对。 我们的方法基于 Lanczos 和 Arnoldi 类型的迭代。 我们证明，使用一个适当定义的预处理器，可能只需要几次迭代就可以很好地估计出在规定区间内的特征值个数。 我们还表明，所提出的预处理方案所需的迭代次数与矩阵的大小和条件数无关。 这些方法的有效性在源自密度泛函理论电子结构计算的几个问题上得到了说明。 显示英文摘要

摘要： We describe preconditioned iterative methods for estimating the number of eigenvalues of a Hermitian matrix within a given interval. Such estimation is useful in a number of applications.In particular, it can be used to develop an efficient spectrum-slicing strategy to compute many eigenpairs of a Hermitian matrix. Our method is based on the Lanczos- and Arnoldi-type of iterations. We show that with a properly defined preconditioner, only a few iterations may be needed to obtain a good estimate of the number of eigenvalues within a prescribed interval. We also demonstrate that the number of iterations required by the proposed preconditioned schemes is independent of the size and condition number of the matrix. The efficiency of the methods is illustrated on several problems arising from density functional theory based electronic structure calculations.