A note on the error analysis of classical Gram–Schmidt |
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Authors: | Alicja Smoktunowicz Jesse L Barlow Julien Langou |
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Institution: | (1) Faculty of Mathematics and Information Science, Warsaw University of Technology, Pl. Politechniki 1, Warsaw, 00-661, Poland;(2) Department of Computer Science and Engineering, The Pennsylvania State University, University Park, PA 16802-6822, USA;(3) Department of Mathematics, University of Colorado at Denver and Health Sciences Center, Denver, USA |
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Abstract: | An error analysis result is given for classical Gram–Schmidt factorization of a full rank matrix A into A = QR where Q is left orthogonal (has orthonormal columns) and R is upper triangular. The work presented here shows that the computed R satisfies R
T
R = A
T
A + E where E is an appropriately small backward error, but only if the diagonals of R are computed in a manner similar to Cholesky factorization of the normal equations matrix. At the end of the article, implications for classical Gram–Schmidt with reorthogonalization are noted.A similar result is stated in Giraud et al. (Numer Math 101(1):87–100, 2005). However, for that result to hold, the diagonals of R must be computed in the manner recommended in this work.Jesse Barlow’s research was supported by the National Science Foundation under grant no. CCF-0429481. |
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