This paper introduces a randomized Householder QR factorization (RHQR). This factorization can be used to obtain a well conditioned basis of a set of vectors and thus can be employed in a variety of applications. The RHQR factorization of the input matrix $W$ is equivalent to the standard Householder QR factorization of matrix $\Psi W$, where $\Psi$ is a sketching matrix that can be obtained from any subspace embedding technique. For this reason, the RHQR algorithm can also be reconstructed from the Householder QR factorization of the sketched problem, yielding a single-synchronization randomized QR factorization (reconstructRHQR). In most contexts, left-looking RHQR requires a single synchronization per iteration, with half the computational cost of Householder QR, and a similar cost to Randomized Gram-Schmidt (RGS) overall. We discuss the usage of RHQR factorization in the Arnoldi process and then in GMRES, showing thus how it can be used in Krylov subspace methods to solve systems of linear equations. Based on Charles Sheffield's connection between Householder QR and Modified Gram-Schmidt (MGS), a BLAS2-RGS is also derived. Numerical experiments show that RHQR produces a well conditioned basis whose sketch is numerically orthogonal even for the most difficult inputs, and an accurate factorization. The same results were observed with the high-dimensional operations made in half-precision. The reconstructed RHQR from the HQR factorization of the sketch was stabler than the standard Randomized Cholesky QR. The first version of this work was made available on HAL on the 7th of July 2023 and can be found at https://hal.science/hal-04156310/

翻译：本文提出了一种随机化Householder QR分解（RHQR）。该分解可用于获得一组向量的良态基，因而能够应用于多种场景。输入矩阵$W$的RHQR分解等价于矩阵$\Psi W$的标准Householder QR分解，其中$\Psi$是通过任意子空间嵌入技术获得的草图矩阵。基于此，RHQR算法也可以通过草图问题的Householder QR分解重构，得到单同步随机化QR分解（reconstructRHQR）。在大多数情况下，左向RHQR每次迭代仅需一次同步，计算成本仅为标准Householder QR分解的一半，且总体成本与随机化Gram-Schmidt（RGS）算法相当。本文讨论了RHQR分解在Arnoldi过程中的应用，进而将其用于GMRES算法，展示了如何将RHQR用于Krylov子空间方法求解线性方程组。基于Charles Sheffield揭示的Householder QR与修正Gram-Schmidt（MGS）之间的关系，本文还推导出了BLAS2-RGS算法。数值实验表明，即使面对最棘手的输入，RHQR也能生成具有数值正交草图的良态基，并实现精确分解。半数精度的高维运算也得到了相同的结果。基于草图HQR分解重构的RHQR比标准随机化Cholesky QR更稳定。本工作第一版于2023年7月7日发布于HAL平台，地址为https://hal.science/hal-04156310/