Shifted CholeskyQR for sparse matrices

Abstract: In this work, we focus on Shifted CholeskyQR for sparse matrices, which is widely used in real applications. We introduce a new model for sparse matrices, categorizing them into two types, $T_{1}$ matrices and $T_{2}$ matrices, based on the presence of dense columns. We provide an alternative choice of the shifted item $s$ for Shifted CholeskyQR3 \cite{Shifted} based on the structure and the key element of the input sparse $X$. We do rounding error analysis of Shifted CholeskyQR3 with such an $s$ and show that it is optimal compared to the original one in \cite{New} with proper element-norm conditions (ENCs) for $T_{1}$ matrices, improving the applicability while maintaining numerical stability. Our theoretical analysis utilizes the properties of $[\cdot]{g}$ proposed in \cite{New}, which is the first to build connections between rounding error analysis and sparse matrices. Numerical experiments confirm our findings for $T{1}$ matrices. Additionally, Shifted CholeskyQR3 with the alternative choice $s$ is applicable to $T_{2}$ matrices, which are more ill-conditioned than dense cases. Furthermore, Shifted CholeskyQR3 with our alternative $s$ shows good efficiency for both $T_{1}$ and $T_{2}$ matrices.