Solving Block Low-Rank Linear Systems by LU Factorization is Numerically Stable Nicholas Higham, Théo Mary To cite this version: Nicholas Higham, Théo Mary. Solving Block Low-Rank Linear Systems by LU Factorization is Numerically Stable. IMA Journal of Numerical Analysis, Oxford University Press (OUP), 2021, 10.1093/imanum/drab020. hal-02496325v3 HAL Id: hal-02496325 https://hal.archives-ouvertes.fr/hal-02496325v3 Submitted on 7 Jun 2021 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Solving Block Low-Rank Linear Systems by LU Factorization is Numerically Stabley NICHOLAS J. HIGHAM Department of Mathematics, University of Manchester, Manchester, M13 9PL, UK (
[email protected]) THEO MARY Sorbonne Universite,´ CNRS, LIP6, Paris, France. (
[email protected]) Block low-rank (BLR) matrices possess a blockwise low-rank property that can be exploited to reduce the complexity of numerical linear algebra algorithms. The impact of these low-rank approximations on the numerical stability of the algorithms in floating-point arithmetic has not previously been analyzed. We present rounding error analysis for the solution of a linear system by LU factorization of BLR matri- ces. Assuming that a stable pivoting scheme is used, we prove backward stability: the relative backward error is bounded by a modest constant times e, where the low-rank threshold e is the parameter control- ling the accuracy of the blockwise low-rank approximations.