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.