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：本文针对统计应用（包括回归、滤波和模型选择）中数据结构频繁变化的情况，提出了一系列快速R更新算法。尽管传统的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，其计算负担会变得非常沉重。所提出的算法能够高效地更新R矩阵而无需重新计算Q矩阵，从而显著降低了计算成本。这些算法为高维回归模型提供了可扩展的解决方案，增强了数据密集型领域中大规模统计分析和模型选择的可行性。全面的模拟研究和实际数据应用表明，这些方法在保持精度的同时显著减少了计算时间。深入的讨论突出了快速R更新算法的多功能性，阐明了其在统计学和机器学习广泛模型与应用中的优势。