This work is about rounding error analysis of randomized CholeskyQR-type algorithms for sparse matrices. We often encounter QR factorization of the sparse matrices in many real problems. In this work, we focus on some typical CholeskyQR-type algorithms with matrix sketching, which is a popular randomized technique in recent years. We build a new model of the sparse matrices and provide rounding error analysis of randomized CholeskyQR-type algorithms for the sparse cases with this model. We make comparison between the bounds with different models of sparsity both theoretically and experimentally. Numerical experiments show some new phenomena of randomized CholeskyQR-type algorithms for the sparse cases, which do not occur in the common sparse cases. We also test the applicability, accuracy, efficiency and robustness of randomized CholeskyQR-type algorithms for sparse matrices.

翻译：本研究针对稀疏矩阵的随机化CholeskyQR类算法进行舍入误差分析。在许多实际问题中，我们常遇到稀疏矩阵的QR分解。本文重点研究结合矩阵素描技术的典型CholeskyQR类算法，该随机化技术近年来备受关注。我们建立了稀疏矩阵的新模型，并基于该模型对稀疏情形下的随机化CholeskyQR类算法进行舍入误差分析。从理论和实验两个维度比较了不同稀疏度模型下的误差界。数值实验揭示了稀疏情形下随机化CholeskyQR类算法的新现象，这些现象在常规稀疏情形中并未出现。我们还测试了随机化CholeskyQR类算法处理稀疏矩阵的适用性、精度、效率与鲁棒性。