Gradient-type iterative methods for solving Hermitian eigenvalue problems can be accelerated by using preconditioning and deflation techniques. A preconditioned steepest descent iteration with implicit deflation (PSD-id) is one of such methods. The convergence behavior of the PSD-id is recently investigated based on the pioneering work of Samokish on the preconditioned steepest descent method (PSD). The resulting non-asymptotic estimates indicate a superlinear convergence of the PSD-id under strong assumptions on the initial guess. The present paper utilizes an alternative convergence analysis of the PSD by Neymeyr under much weaker assumptions. We embed Neymeyr's approach into the analysis of the PSD-id using a restricted formulation of the PSD-id. More importantly, we extend the new convergence analysis of the PSD-id to a practically preferred block version of the PSD-id, or BPSD-id, and show the cluster robustness of the BPSD-id. Numerical examples are provided to validate the theoretical estimates.

翻译：使用先决条件和通货紧缩技术,可以加速加速采用梯度式迭代方法解决Hermitian egenvaly问题; 一种以隐含通缩(PSD-id)为先决条件的急剧下降迭代是这种方法之一; 最近根据Samokish关于最隐含性下降方法(PSD)的开拓性工作,调查了私营部门司的趋同行为; 由此得出的非被动估计表明私营部门司在初步猜测的强烈假设下具有超线性趋同; 本文利用Neymeyr对私营部门司的较弱假设进行的另一种趋同性分析。 我们用私营部门司的有限公式将Neymeyyr的方法纳入私营部门司的趋同分析。 更重要的是,我们将私营部门司对私营部门司的新趋同性分析扩大到私营部门司(PSD-id)的一个实际首选的区块版,或BPSD-id, 显示私营部门司的集群集稳健性。 提供了数字实例,以证实理论估计。