This paper introduces a randomized Householder QR factorization (RHQR). This factorization can be used to obtain a well conditioned basis of a vector space 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 factorization can also be reconstructed from the Householder QR factorization of the sketched problem, yielding a single-synchronization randomized QR factorization (recRHQR). 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. A finite precision analysis shows that, under mild probabilistic assumptions, the RHQR factorization of the input matrix $W$ inherits the stability of the Householder QR factorization, producing a well-conditioned basis and a columnwise backward stable factorization, all independently of the condition number of the input $W$, and with the accuracy of the sketching step. We study the subsampled randomized Hadamard transform (SRHT) as a very stable sketching technique. Numerical experiments show that RHQR produces a well conditioned basis whose sketch is numerically orthogonal and an accurate factorization, even for the most difficult inputs and with high-dimensional operations made in half-precision.

翻译：本文提出了一种随机化Householder QR分解方法（RHQR）。该分解可用于获取向量空间的一个良条件基，因此可应用于多种场景。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$\Psi$为可通过任意子空间嵌入技术获得的素描矩阵。基于此特性，RHQR分解也可通过素描化问题的Householder QR分解重构得到，从而形成单同步随机化QR分解（recRHQR）。在多数情况下，左视RHQR每次迭代仅需一次同步，其计算成本仅为Householder QR的一半，总体成本与随机化Gram-Schmidt方法（RGS）相当。本文探讨了RHQR分解在Arnoldi过程及GMRES中的应用，展示了其在Krylov子空间方法中求解线性方程组的潜力。基于Charles Sheffield建立的Householder QR与修正Gram-Schmidt（MGS）之间的关联，本文还推导出BLAS2-RGS算法。有限精度分析表明，在温和的概率假设下，输入矩阵$W$的RHQR分解继承了Householder QR分解的稳定性，能够产生良条件基和列向向后稳定分解，且这些特性与输入矩阵$W$的条件数无关，仅取决于素描步骤的精度。本文研究了子采样随机化Hadamard变换（SRHT）作为一种高稳定性素描技术。数值实验表明，即使对于最困难的输入矩阵及采用半精度计算的高维运算，RHQR仍能生成良条件基（其素描保持数值正交性）并提供精确的分解结果。