We consider numerical methods for computing eigenvalues located in the interior part of the spectrum of a large symmetric matrix. For such difficult eigenvalue problems, an effective solution is to use the Harmonic Ritz pairs in projection methods because the error bounds on the Harmonic Ritz pairs are well studied. In this paper, we prove global convergence of the iterative projection methods with the Harmonic Ritz pairs in an abstract form, where the standard restart strategy is employed. To this end, we reformulate the existing convergence proof of the Ritz pairs to be successfully applied to the Harmonic Ritz pairs with the inexact linear system solvers. Our main theorem obtained by the above convergence analysis shows important features concerning the global convergence of the Harmonic Ritz pairs.

中文翻译：

---

具有对称特征值问题的重新启动策略的谐波 Ritz 对迭代投影方法的收敛证明

我们考虑计算位于大型对称矩阵谱内部的特征值的数值方法。对于此类困难的特征值问题，有效的解决方案是在投影方法中使用 Harmonic Ritz 对，因为对 Harmonic Ritz 对的误差界限进行了很好的研究。在本文中，我们以抽象形式证明了迭代投影方法与 Harmonic Ritz 对的全局收敛性，其中采用了标准的重启策略。为此，我们重新制定了 Ritz 对的现有收敛证明，以成功应用于具有不精确线性系统求解器的 Harmonic Ritz 对。我们通过上述收敛分析得到的主要定理显示了关于谐波 Ritz 对全局收敛的重要特征。