@article {17693,
	title = {Block Gram{\textendash}Schmidt Orthogonalization},
	journal = {SIAM Journal on Scientific Computing},
	volume = {31},
	year = {2008},
	month = {2008///},
	pages = {761 - 775},
	abstract = {The classical Gram{\textendash}Schmidt algorithm for computing the QR factorization of a matrix $X$ requires at least one pass over the current orthogonalized matrix $Q$ as each column of $X$ is added to the factorization. When $Q$ becomes so large that it must be maintained on a backing store, each pass involves the costly transfer of data from the backing store to main memory. However, if one orthogonalizes the columns of $X$ in blocks of $m$ columns, the number of passes is reduced by a factor of $1/m$. Moreover, matrix-vector products are converted into matrix-matrix products, allowing level-3 BLAS cache performance. In this paper we derive such a block algorithm and give some experimental results that suggest it can be quite effective for large scale problems, even when the matrix $X$ is rank degenerate.},
	keywords = {blocked algorithm, GramSchmidt algorithm, orthogonalization, QR factorization},
	doi = {10.1137/070682563},
	url = {http://link.aip.org/link/?SCE/31/761/1},
	author = {Stewart, G.W.}
}
@article {17698,
	title = {Error Analysis of the Quasi-Gram{\textendash}Schmidt Algorithm},
	journal = {SIAM Journal on Matrix Analysis and Applications},
	volume = {27},
	year = {2005},
	month = {2005///},
	pages = {493 - 506},
	abstract = {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.},
	keywords = {Gram{\textendash}Schmidt algorithm, orthogonalization, QR factorization, rounding-error analysis, sparse matrix},
	doi = {10.1137/040607794},
	url = {http://link.aip.org/link/?SML/27/493/1},
	author = {Stewart, G.W.}
}
