Updating a linear least squares solution can be critical for near real-time signalprocessing applications. The Greville algorithm proposes a simple formula for updating the pseudoinverse of a matrix A $\in$ R nxm with rank r. In this paper, we explicitly derive a similar formula by maintaining a general rank factorization, which we call rank-Greville. Based on this formula, we implemented a recursive least squares algorithm exploiting the rank-deficiency of A, achieving the update of the minimum-norm least-squares solution in O(mr) operations and, therefore, solving the linear least-squares problem from scratch in O(nmr) operations. We empirically confirmed that this algorithm displays a better asymptotic time complexity than LAPACK solvers for rank-deficient matrices. The numerical stability of rank-Greville was found to be comparable to Cholesky-based solvers. Nonetheless, our implementation supports exact numerical representations of rationals, due to its remarkable algebraic simplicity.

中文翻译：

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秩亏矩阵的高效递归最小二乘求解器

更新线性最小二乘解对于近乎实时的信号处理应用至关重要。Greville 算法提出了一个简单的公式来更新秩为 r 的矩阵 A $\in$ R nxm 的伪逆。在本文中，我们通过保持一般秩分解明确推导出一个类似的公式，我们称之为 rank-Greville。基于这个公式，我们实现了一个递归最小二乘算法，利用了 A 的秩不足，实现了 O(mr) 操作中的最小范数最小二乘解的更新，因此，解决了线性最小二乘问题在 O(nmr) 操作中划伤。我们凭经验证实，对于秩亏矩阵，该算法显示出比 LAPACK 求解器更好的渐近时间复杂度。发现 rank-Greville 的数值稳定性与基于 Cholesky 的求解器相当。尽管如此，由于其显着的代数简单性，我们的实现支持有理数的精确数值表示。