CholeskyQR-type algorithms are very popular in both academia and industry in recent years. It could make a balance between the computational cost, accuracy and speed. CholeskyQR2 provides numerical stability of orthogonality and Shifted CholeskyQR3 deals with problems regarding ill-conditioned matrices. 3C algorithm is applicable for sparse matrices. However, the overestimation of the error matrices in the previous works influences the sufficient conditions for these algorithms. Particularly, it leads to a conservative shifted item in Shifted CholeskyQR3 and 3C, which may greatly influence the properties of the algorithms. In this work, we consider the randomized methods and utilize the model of probabilistic error analysis in \cite{New} to do rounding error analysis for CholeskyQR-type algorithms. We combine the theoretical analysis with the $g$-norm defined in \cite{Columns}. Our analysis could provide a smaller shifted item for Shifted CholeskyQR3 and could improve the orthogonality of our 3C algorithm for dense matrices. Numerical experiments in the final section shows that our improvements with randomized methods do have some advantages compared with the original algorithms.

翻译：近年来，CholeskyQR类算法在学术界和工业界广受欢迎。该算法能在计算成本、精度和速度之间取得良好平衡。CholeskyQR2保证了正交性的数值稳定性，Shifted CholeskyQR3则能处理病态矩阵问题。3C算法适用于稀疏矩阵场景。然而，现有研究中对误差矩阵的高估影响了这些算法的充分条件。特别地，这导致Shifted CholeskyQR3和3C算法中的偏移项设置过于保守，可能显著影响算法性能。本研究采用随机化方法，基于\cite{New}中的概率误差分析模型对CholeskyQR类算法进行舍入误差分析。我们将理论分析与\cite{Columns}中定义的$g$-范数相结合。通过分析可为Shifted CholeskyQR3提供更小的偏移项，并提升3C算法在稠密矩阵场景下的正交性。最终节的数值实验表明，采用随机化方法的改进方案相较于原始算法确实具有优势。