Chebyshev filtered subspace iteration is a well-known algorithm for the solution of (symmetric/Hermitian) algebraic eigenproblems which has been implemented in several application codes~\cite{Kronik:2006ff, abinit:2020} or in stand alone libraries~\cite{ChASE}. An essential part of the algorithm is the QR-factorization of the array of vectors spanning the active subspace that have been filtered by the Chebyshev filter. Typically such an array has an a-priori unknown high condition number that directly influences the choice of QR-factorization algorithm. In this work we show how such condition number can be bound from above with precise and inexpensive estimates. We then proceed to use these estimates to implement a mechanism for the choice of QR-factorization in the ChASE library. We show how such mechanism enhance the performance of the library without compromising on its accuracy.

翻译：切比雪夫滤波子空间迭代是求解（对称/厄米特）代数特征值问题的经典算法，已在多个应用代码~\cite{Kronik:2006ff, abinit:2020} 及独立库~\cite{ChASE} 中实现。该算法的核心环节是对经切比雪夫滤波器作用后的活动子空间向量组进行QR分解。此类向量组通常具有先验未知的高条件数，直接影响QR分解算法的选择。本文提出一种精确且低计算代价的上界估计方法，用以界定该条件数的范围。基于此估计，我们在ChASE库中实现了QR分解算法的自适应选择机制。实验表明，该机制在保证计算精度的同时显著提升了库的整体性能。