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Reorthogonalization and stable algorithms for updating the Gram-Schmidt $ QR$ factorization


Authors: J. W. Daniel, W. B. Gragg, L. Kaufman and G. W. Stewart
Journal: Math. Comp. 30 (1976), 772-795
MSC: Primary 65F25
DOI: https://doi.org/10.1090/S0025-5718-1976-0431641-8
MathSciNet review: 0431641
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Abstract: Numerically stable algorithms are given for updating the Gram-Schmidt QR factorization of an $ m \times n$ matrix $ A\;(m \geqslant n)$ when A is modified by a matrix of rank one, or when a row or column is inserted or deleted. The algorithms require $ O(mn)$ operations per update, and are based on the use of elementary two-by-two reflection matrices and the Gram-Schmidt process with reorthogonalization. An error analysis of the reorthogonalization process provides rigorous justification for the corresponding ALGOL procedures.


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DOI: https://doi.org/10.1090/S0025-5718-1976-0431641-8
Article copyright: © Copyright 1976 American Mathematical Society

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