Probabilistic error analysis of CholeskyQR based on columns

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.