This paper introduces fast R updating algorithms designed for statistical applications, including regression, filtering, and model selection, where data structures change frequently. Although traditional QR decomposition is essential for matrix operations, it becomes computationally intensive when dynamically updating the design matrix in statistical models. The proposed algorithms efficiently update the R matrix without recalculating Q, significantly reducing computational costs. These algorithms provide a scalable solution for high-dimensional regression models, enhancing the feasibility of large-scale statistical analyses and model selection in data-intensive fields. Comprehensive simulation studies and real-world data applications reveal that the methods significantly reduce computational time while preserving accuracy. An extensive discussion highlights the versatility of fast R updating algorithms, illustrating their benefits across a wide range of models and applications in statistics and machine learning.

翻译：本文介绍了专为统计应用设计的快速R更新算法，适用于回归、滤波和模型选择等数据结构频繁变化的场景。尽管传统QR分解在矩阵运算中至关重要，但在统计模型中动态更新设计矩阵时，其计算成本会急剧增加。所提出的算法能在不重新计算Q矩阵的情况下高效更新R矩阵，显著降低了计算开销。这些算法为高维回归模型提供了可扩展的解决方案，提升了数据密集型领域大规模统计分析与模型选择的可行性。综合模拟研究和实际数据应用表明，该方法在保持精度的同时显著减少了计算时间。深入讨论揭示了快速R更新算法的多功能性，阐明了其在统计学与机器学习各类模型和应用中的广泛效益。