We a present and analyze rpCholesky-QR, a randomized preconditioned Cholesky-QR algorithm for computing the thin QR factorization of real mxn matrices with rank n. rpCholesky-QR has a low orthogonalization error, a residual on the order of machine precision, and does not break down for highly singular matrices. We derive rigorous and interpretable two-norm perturbation bounds for rpCholesky-QR that require a minimum of assumptions. Numerical experiments corroborate the accuracy of rpCholesky-QR for preconditioners sampled from as few as 3n rows, and illustrate that the two-norm deviation from orthonormality increases with only the condition number of the preconditioned matrix, rather than its square -- even if the original matrix is numerically singular.

翻译：本文提出并分析了一种用于计算秩为n的实m×n矩阵薄QR分解的随机化预条件Cholesky-QR算法——rpCholesky-QR。该算法具有较低的正交化误差、机器精度量级的残差，且在处理高度奇异矩阵时不会失效。我们推导了rpCholesky-QR严格且可解释的二范数扰动界，该界仅需最小化假设条件。数值实验证实，即使仅从3n行采样预条件子，rpCholesky-QR仍能保持计算精度，并证明正交归一性的二范数偏差仅随预条件矩阵的条件数增长（而非其平方增长）——即使原始矩阵是数值奇异的。