Efficient recursive least squares solver for rank-deficient matrices

Ruben Staub and Stephan N. Steinmann

Applied Mathematics and Computation, 2021, vol. 399, issue C

Abstract: Updating a linear least-squares solution can be critical for near real-time signal-processing applications. The Greville algorithm proposes a simple formula for updating the pseudoinverse of a matrix A∈Rn×m 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.

Keywords: Moore-Penrose pseudoinverse; Generalized inverse; Recursive least-squares; Rank-deficient linear systems (search for similar items in EconPapers)
Date: 2021
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Persistent link: https://EconPapers.repec.org/RePEc:eee:apmaco:v:399:y:2021:i:c:s0096300321000448

DOI: 10.1016/j.amc.2021.125996

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