This paper develops and analyzes a new algorithm for QR decomposition with column pivoting (QRCP) of rectangular matrices with large row counts. The algorithm combines methods from randomized numerical linear algebra in a particularly careful way in order to accelerate both pivot decisions for the input matrix and the process of decomposing the pivoted matrix into the QR form. The source of the latter acceleration is a 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 granted probabilistic assumptions of the sketching operator. An implementation of the proposed algorithm is described and made available inside the open-source RandLAPACK library, which itself relies on RandBLAS - also available in open-source format. 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, which lacks the strong rank-revealing properties of the proposed method.

翻译：摘要：本文提出并分析了一种针对行数众多的矩形矩阵的列选主元QR分解（QRCP）新算法。该算法以极为审慎的方式融合了随机数值线性代数方法，旨在同时加速输入矩阵的主元选择过程以及将选主元后矩阵分解为QR形式的过程。后者加速的来源在于随机预处理与CholeskyQR技术的运用。本文分别在精确算术与有限精度算术下提供了全面分析，以刻画算法在概率性素描算子假设下的秩揭示特性及其数值稳定性。文中描述了所提算法的实现方案，并将其集成至开源库RandLAPACK中（该库本身依赖于同样开源的RandBLAS）。在Intel Xeon Gold 6248R CPU上的实验表明，该实现相对LAPACK标准QRCP函数可实现数量级加速，且性能与专用于高矩阵无选主元QR的算法相当，而后者缺乏本方法所具有的强秩揭示特性。