A note on the error analysis of classical Gram-Schmidt

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

2006-06-11

arxiv.org/abs/math/0606258v2

Numerische Mathematik, 105(2):299-313, December 2006

Mathematics

Numerical Analysis

12 pages This v2. v1 (from 2006) has not the biliographical reference set (at all). This is the only modification between v1 a

Scientific paper

10.1007/s00211-006-0042-1

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 $\normal{R}=\normal{A}+E$ 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. A similar result is stated in [Giraud at 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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