Randomized Householder QR

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/