Error Analysis of the Quasi-Gram--Schmidt Algorithm

Let the $n{\times}p$ $(n\geq p)$ matrix $X$ have the QR~factorization$X = QR$, where $R$ is an upper triangular matrix of order $p$ and $Q$
is orthonormal. This widely used decomposition has the drawback that
$Q$ is not generally sparse even when $X$ is. One cure is to discard
$Q$ retaining only $X$ and $R$. Products like $a = Q\trp y = R\itp
X\trp y$ can then be formed by computing $b = X\trp y$ and solving the
system $R\trp a = b$. This approach can be used to modify the
Gram--Schmidt algorithm for computing $Q$ and $R$ to compute $R$
without forming $Q$ or altering $X$. Unfortunately, this
quasi-Gram--Schmidt algorithm can produce inaccurate results. In this
paper it is shown that with reorthogonalization the inaccuracies are
bounded under certain natural conditions.
(UMIACS-TR-2004-17)