Abstract
In this work, we focus on Shifted CholeskyQR (SCholeskyQR) for sparse matrices. For SCholeskyQR, the choice of the shifted item $s$ will influence the property of the algorithm. Therefore, we provide a new shifted item $s$ for Shifted CholeskyQR3 (SCholeskyQR3) based on the number of non-zero elements (nnz) and the element with the largest absolute value of the input sparse $X \in \mathbb{R}^{m\times n}$ with $m \ge n$. 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. We do rounding error analysis of SCholeskyQR3 with the new $s$ and prove that this $s$ can improve the applicability and accuracy of the algorithm for the sparse cases. Numerical experiments demonstrate the advantage of SCholeskyQR3 with our alternative choice of $s$ in both applicability and accuracy over the case with the original one, together with similar efficiency to that of the thin HouseholderQR in many cases. 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.