Randomized LU-Householder CholeskyQR

In this work, we develop randomized LU-Householder CholeskyQR (rLHC) for QR factorization of the tall-skinny matrices, consisting of SLHC3 with single-sketching and SSLHC3 with multi-sketching. Similar to LU-CholeskyQR2 (LUC2), they do not require a condition of $\kappa_{2}(X)$ for the input matrix $X \in \mathbb{R}^{m\times n}$, which ensures the applicability of the algorithms and is distinguished from many CholeskyQR-type algorithms. To address the issue of numerical breakdown of LUC2 when the $L$-factor from LU factorization is ill-conditioned, we employ HouseholderQR to generate the upper-triangular factor alternatively together the latest matrix sketching to guarantee the efficiency. We provide rounding error analysis of our new algorithms and show their numerical stability. Numerical experiments demonstrate the better applicability of SLHC3 and SSLHC3 compared to the existing algorithms while maintaining good accuracy and robustness. Regarding efficiency, SLHC3 and SSLHC3 are at the similar level as that of LUC2. When $m=\mathcal{O}(n^{2})$ for $X$, SSLHC3 with multi-sketching is more efficient than SLHC3 with single-sketching. SLHC3 and SSLHC3 balance the applicability, efficiency, accuracy and robustness, becoming outstanding algorithms for QR factorization.