Probabilistic error analysis of CholeskyQR based on columns

Abstract: In this work, we utilize the randomized models presented in \cite{New} and do probabilistic error analysis of CholeskyQR2 and Shifted CholeskyQR3. We integrate the theoretical analysis with $[\cdot]_{g}$ defined in \cite{Columns}, providing sharper upper bounds of accuracy and better sufficient conditions for CholeskyQR2 and Shifted CholeskyQR3 along with the corresponding probabilities. Moreover, a probabilistic shifted item $s$ for Shifted CholeskyQR3 is received, improving the applicability of the algorithm while maintaining numerical stability. Numerical experiments confirm our findings and show that such a probabilistic $s$ has good robustness in ill-conditioned cases.