TY - JOUR

T1 - A note on the error analysis of classical Gram-Schmidt

AU - Smoktunowicz, Alicja

AU - Barlow, Jesse L.

AU - Langou, Julien

N1 - Copyright:
Copyright 2006 Elsevier B.V., All rights reserved.

PY - 2006/12

Y1 - 2006/12

N2 - 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.

AB - 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.

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U2 - 10.1007/s00211-006-0042-1

DO - 10.1007/s00211-006-0042-1

M3 - Article

AN - SCOPUS:33751159603

SN - 0029-599X

VL - 105

SP - 299

EP - 313

JO - Numerische Mathematik

JF - Numerische Mathematik

IS - 2

ER -