In this work, we focus on improving LU-CholeskyQR2 \cite{LUChol}. Compared to other deterministic and randomized CholeskyQR-type algorithms, it does not require a sufficient condition on $\kappa_{2}(X)$ for the input tall-skinny matrix $X$, which ensures the algorithm's safety in most real-world applications. However, the Cholesky factorization step may break down when the $L$-factor after the LU factorization of $X$ is ill-conditioned. To address this, we construct a new algorithm, LU-Householder CholeskyQR2 (LHC2), which uses HouseholderQR to generate the upper-triangular factor, thereby avoiding numerical breakdown. Moreover, we utilize the latest sketching techniques to develop randomized versions of LHC: SLHC and SSLHC. We provide a rounding error analysis for these new algorithms. Numerical experiments demonstrate that our three new algorithms have better applicability and can handle a wider range of matrices compared to LU-CholeskyQR2. With the sketching technique, our randomized algorithms, SLHC2 and SSLHC3, show significant acceleration over LHC2. Additionally, SSLHC3, which employs multi-sketching, is more efficient than SLHC2 and exhibits better numerical stability. It is also robust as a randomized algorithm.

翻译：本文工作中，我们专注于改进LU-CholeskyQR2算法。与其他确定性和随机化CholeskyQR类算法相比，该算法不要求输入的高瘦矩阵X满足κ₂(X)的充分条件，这确保了算法在大多数实际应用中的安全性。然而，当X进行LU分解后的L因子病态时，Cholesky分解步骤可能失效。为解决此问题，我们构建了一种新算法——LU-Householder CholeskyQR2（LHC2），该算法利用HouseholderQR生成上三角因子，从而避免数值失效。此外，我们采用最新的草图技术开发了LHC的随机化版本：SLHC与SSLHC。我们为这些新算法提供了舍入误差分析。数值实验表明，与LU-CholeskyQR2相比，我们提出的三种新算法具有更好的适用性，并能处理更广泛的矩阵类型。借助草图技术，我们的随机化算法SLHC2和SSLHC3相比LHC2展现出显著的加速效果。特别地，采用多重草图技术的SSLHC3比SLHC2效率更高，且表现出更优的数值稳定性。作为随机化算法，SSLHC3亦具备良好的鲁棒性。