This paper develops and analyzes an algorithm for QR decomposition with column pivoting (QRCP) of tall matrices. The algorithm uses methods from randomized numerical linear algebra in a particularly careful way, to accelerate both pivot decisions for the input matrix and the process of decomposing the pivoted matrix via QR. The source of the latter acceleration is the use of randomized preconditioning and CholeskyQR. Comprehensive analysis is provided in both exact and finite-precision arithmetic to characterize the algorithm's rank-revealing properties and its numerical stability. An implementation of the described algorithm is made available under the open-source RandLAPACK library, which itself relies on RandBLAS. Experiments with this implementation on an Intel Xeon Gold 6248R CPU demonstrate order-of-magnitude speedups relative to LAPACK's standard function for QRCP, and comparable performance to a specialized algorithm for unpivoted QR of tall matrices.

翻译：本文提出并分析了一种面向高瘦矩阵的列选主元QR分解（QRCP）算法。该算法以极为审慎的方式运用随机数值线性代数方法，在输入矩阵的主元决策和通过QR分解实现主元矩阵分解这两个环节上实现加速。后者的加速源于随机预处理与CholeskyQR技术的结合。本文在精确算术和有限精度算术下进行了全面分析，以刻画算法的秩揭示特性及其数值稳定性。所述算法的实现已基于开源库RandLAPACK发布，该库自身依赖于RandBLAS。在Intel Xeon Gold 6248R CPU上进行的实验表明，该实现相对于LAPACK标准QRCP函数实现了数量级加速，且性能与面向高瘦矩阵的无选主元QR专用算法相当。