In this work, we focus on Shifted CholeskyQR (SCholeskyQR) for sparse matrices. We provide a new shifted item $s$ for Shifted CholeskyQR3 (SCholeskyQR3) based on the number of non-zero elements (nnze) and the element with the largest absolute value of the input sparse $X \in \mathbb{R}^{m\times n}$ with $m \ge n$. We do rounding error analysis of SCholeskyQR3 with such an $s$ and show that SCholeskyQR3 is accurate in this case. Therefore, an alternative choice of $s$ can be taken for SCholeskyQR3 with the comparison between our new $s$ and the $s$ shown in the previous work when the input $X$ is sparse, improving the applicability and residual of the algorithm for the ill-conditioned cases. Numerical experiments demonstrate the advantage of SCholeskyQR3 with our alternative choice of $s$ in both applicablity and accuracy over the case with the original $s$, together with the same level of efficiency. This work is also the first to build connections between sparsity and numerical algorithms with detailed rounding error analysis to the best of our knowledge.

翻译：本文聚焦于稀疏矩阵的移位CholeskyQR（SCholeskyQR）方法。针对输入稀疏矩阵 $X \in \mathbb{R}^{m\times n}$（其中 $m \ge n$），我们基于其非零元素数量（nnze）及元素绝对值的最大值，为Shifted CholeskyQR3（SCholeskyQR3）提出了一种新的移位项 $s$。我们对采用该 $s$ 的SCholeskyQR3进行了舍入误差分析，并证明在此情况下SCholeskyQR3具有数值准确性。因此，当输入矩阵 $X$ 为稀疏矩阵时，可通过对比本文提出的新 $s$ 与既有研究中给出的 $s$，为SCholeskyQR3提供更优的移位项选择，从而提升算法在病态情况下的适用性与残差性能。数值实验表明，采用本文替代 $s$ 的SCholeskyQR3在保持同等效率的同时，其适用性与精度均优于采用原始 $s$ 的情况。据我们所知，本研究首次通过详尽的舍入误差分析，建立了稀疏性与数值算法之间的理论联系。