CholeskyQR is an efficient algorithm for QR factorization with several advantages compared with orhter algorithms. In order to improve its orthogonality, CholeskyQR2 is developed \cite{2014}\cite{error}. To deal with ill-conditioned matrices, a shifted item $s$ is introduced and we have Shifted CholeskyQR3 \cite{Shifted}. In many problems in the industry, QR factorization for sparse matrices is very common, especially for some sparse matrices with special structures. In this work, we discuss the property of CholeskyQR-type algorithms for sparse matrices. We introduce new definitions for the input sparse matrix $X$ and divide them into two types based on column properties. We provide better sufficient conditions for $\kappa_{2}(X)$ and better shifted item $s$ for CholeskyQR-type algorithms under certain element-norm conditiones(ENCs) compared with the original ones in \cite{Shifted}\cite{error}, together with an alternative error analysis for the algorithm. The steps of analysis utilize the properties of the $g$-norm of the matrix which is given in the previous work. Moreover, a new three-step CholeskyQR-type algorithm with two shifted items called 3C is developed for sparse matrices with good orthogonality. We do numerical experiments with some typical real examples to show the advantages of improved algorithms compared with the original ones in the previous works.

翻译：CholeskyQR是一种高效的QR分解算法，相较于其他算法具有多项优势。为提升其正交性，研究者开发了CholeskyQR2算法（参见文献\cite{2014}\cite{error}）。针对病态矩阵的处理，通过引入偏移项$s$，形成了Shifted CholeskyQR3算法（参见文献\cite{Shifted}）。在工业领域的诸多问题中，稀疏矩阵的QR分解极为常见，尤其针对某些具有特殊结构的稀疏矩阵。本文探讨了CholeskyQR类算法在稀疏矩阵上的性质。我们为输入稀疏矩阵$X$提出了新的定义，并根据列特性将其划分为两种类型。在特定元素范数条件（ENCs）下，相较于文献\cite{Shifted}\cite{error}中的原始条件，我们为$\kappa_{2}(X)$提供了更优的充分条件，并为CholeskyQR类算法给出了更佳的偏移项$s$，同时提出了该算法的替代误差分析。分析步骤运用了先前工作中提出的矩阵$g$范数性质。此外，针对稀疏矩阵我们开发了一种包含两个偏移项的新型三步CholeskyQR类算法（称为3C），该算法具有良好的正交性。我们通过典型实际算例进行数值实验，展示了改进算法相较于先前工作中原始算法的优势。