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和位移CholeskyQR3是计算高瘦矩阵QR分解的两种先进算法，因为它们在现代计算机架构上能实现高性能。然而，为保证稳定性，在某些应用中，CholeskyQR2对需分解矩阵的条件数存在严格的限制。位移CholeskyQR3虽稳定，但其计算与通信开销比CholeskyQR2高出50%。本文提出并分析了一种称为随机化Householder-Cholesky（\texttt{rand\_cholQR}）的随机化QR算法。通过使用一个或两个随机草图矩阵，证明了该算法以高概率保证其正交性误差受限于单位舍入误差量级的常数，因此其稳定性与位移CholeskyQR3相当。在NVIDIA A100 GPU上对\texttt{rand\_cholQR}的性能评估表明，对于高瘦矩阵，采用多重草图矩阵的\texttt{rand\_cholQR}速度几乎与CholeskyQR2相当，甚至在某些情况下更快。因此，相较于CholeskyQR2，\texttt{rand\_cholQR}具有更优的稳定性，且几乎不增加计算或内存开销，是从理论到实践均更优越的算法。