Analysis of RCLUPPr: stability, robustness and acceleration

Abstract

In this work, we present a comprehensive rounding error analysis for RCLUPPr, a recently developed randomized CholeskyQR-type algorithm introduced in \cite{RCLUPP}. This method performs LU factorization with partial pivoting (LUPP faztorization) directly on a full-rank tall-skinny matrix $X \in \mathbb{R}^{m \times n}$. Unlike RCLUPP in \cite{RCLUPP}, which applies matrix sketching prior to LUPP factorization, RCLUPPr deploys LUPP as an initial preconditioning step to significantly mitigate the propagation of error significantly. Our rigorous analysis demonstrates that RCLUPPr exhibits superior robustness when applied to the ill-conditioned matrices compared to other CholeskyQR-type algorithms in the high precision. Furthermore, we develop some practical strategies in the implementation to accelerate RCLUPPr. Extensive numerical experiments on the real-world problems validate our theoretical findings, showcasing the robustness and the efficiency of RCLUPPr across the single, double and the mixed-precision frameworks.