Among all the deterministic CholeskyQR-type algorithms, Shifted CholeskyQR3 is specifically designed to address the QR factorization of ill-conditioned matrices. This algorithm introduces a shift parameter $s$ to prevent failure during the initial Cholesky factorization step, making the choice of this parameter critical for the algorithm's effectiveness. Our goal is to identify a smaller $s$ compared to the traditional selection based on $\norm{X}_{2}$. In this research, we propose a new matrix norm called the $g$-norm, which is based on the column properties of $X$. This norm allows us to obtain a reduced shift parameter $s$ for the Shifted CholeskyQR3 algorithm, thereby improving the sufficient condition of $\kappa_{2}(X)$ for this method. We provide rigorous proofs of orthogonality and residuals for the improved algorithm using our proposed $s$. Numerical experiments confirm the enhanced numerical stability of orthogonality and residuals with the reduced $s$. We find that Shifted CholeskyQR3 can effectively handle ill-conditioned $X$ with a larger $\kappa_{2}(X)$ when using our reduced $s$ compared to the original $s$. Furthermore, we compare CPU times with other algorithms to assess performance improvements.

翻译：在所有确定性CholeskyQR类算法中，移位CholeskyQR3专门设计用于处理病态矩阵的QR分解。该算法引入移位参数$s$以避免初始Cholesky分解步骤失败，使得该参数的选择对算法效能至关重要。我们的目标在于找到比传统基于$\norm{X}_{2}$的选择更小的$s$。本研究提出了一种基于$X$列特性的新矩阵范数——$g$-范数，该范数使我们能为移位CholeskyQR3算法获得更小的移位参数$s$，从而改进该方法对$\kappa_{2}(X)$的充分条件。我们为采用所提$s$值的改进算法提供了正交性和残差的严格证明。数值实验证实了采用更小$s$值时正交性与残差数值稳定性的提升。研究发现，相较于原始$s$值，采用我们提出的更小$s$值时，移位CholeskyQR3能有效处理具有更大$\kappa_{2}(X)$的病态矩阵$X$。此外，我们通过与其他算法的CPU时间对比来评估性能改进。