CholeskyQR2 and shifted CholeskyQR3 are two state-of-the-art algorithms for computing tall-and-skinny QR factorizations since they attain high performance on current computer architectures. However, to guarantee stability, for some applications, CholeskyQR2 faces a prohibitive restriction on the condition number of the underlying matrix to factorize. Shifted CholeskyQR3 is stable but has $50\%$ more computational and communication costs than CholeskyQR2. In this paper, a randomized QR algorithm called Randomized Householder-Cholesky (\texttt{rand\_cholQR}) is proposed and analyzed. Using one or two random sketch matrices, it is proved that with high probability, its orthogonality error is bounded by a constant of the order of unit roundoff for any numerically full-rank matrix, and hence it is as stable as shifted CholeskyQR3. An evaluation of the performance of \texttt{rand\_cholQR} on a NVIDIA A100 GPU demonstrates that for tall-and-skinny matrices, \texttt{rand\_cholQR} with multiple sketch matrices is nearly as fast as, or in some cases faster than, CholeskyQR2. Hence, compared to CholeskyQR2, \texttt{rand\_cholQR} is more stable with almost no extra computational or memory cost, and therefore a superior algorithm both in theory and practice.

翻译：CholeskyQR2和移位CholeskyQR3是当前计算高瘦QR分解的两种最先进算法，因其在现代计算机架构上具有高性能。然而，为保证稳定性，在某些应用中，CholeskyQR2对待分解矩阵的条件数存在严格的限制。移位CholeskyQR3是稳定的，但其计算和通信成本比CholeskyQR2高出50%。本文提出并分析一种名为随机Householder-Cholesky ( \texttt{rand\_cholQR} ) 的随机化QR算法。通过使用一或两个随机草图矩阵，证明对于任意数值满秩矩阵，其正交性误差高概率地被限制在单位舍入量级的常数范围内，因此其稳定性与移位CholeskyQR3相当。在NVIDIA A100 GPU上对 \texttt{rand\_cholQR} 性能的评估表明，对于高瘦矩阵，使用多个草图矩阵的 \texttt{rand\_cholQR} 速度几乎与CholeskyQR2相同，甚至在某些情况下更快。因此，与CholeskyQR2相比，\texttt{rand\_cholQR} 在几乎没有额外计算或内存成本的情况下更稳定，从而在理论和实践上均是一种更优的算法。