An improved error analysis of CholeskyQR with the randomized model

This work is about an improved error analysis of CholeskyQR with the randomized model for the tall-skinny $X \in \mathbb{R}^{m\times n}$. Due to the structure of CholeskyQR, we utilize the randomized model in the first step of CholeskyQR with a weak assumption. We receive a better sufficient condition of $\kappa_{2}(X)$ and a tighter upper bound of residual for CholeskyQR2, together with a probabilistic shifted item $s$ for Shifted CholeskyQR3 based on $\norm{X}_{F}$ after improved error analysis. Numerical experiments demonstrate the effectiveness of our new theoretical results. The probabilistic $s$ for Shifted CholeskyQR3 can enhance the applicability of Shifted CholeskyQR3 while maintaining numerical stability. It is also robust enough after numerous experiments.