@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 {19662,
	title = {MCMC Data Association and Sparse Factorization Updating for Real Time Multitarget Tracking with Merged and Multiple Measurements},
	journal = {IEEE Transactions on Pattern Analysis and Machine Intelligence},
	volume = {28},
	year = {2006},
	month = {2006/12//},
	pages = {1960 - 1972},
	abstract = {In several multitarget tracking applications, a target may return more than one measurement per target and interacting targets may return multiple merged measurements between targets. Existing algorithms for tracking and data association, initially applied to radar tracking, do not adequately address these types of measurements. Here, we introduce a probabilistic model for interacting targets that addresses both types of measurements simultaneously. We provide an algorithm for approximate inference in this model using a Markov chain Monte Carlo (MCMC)-based auxiliary variable particle filter. We Rao-Blackwellize the Markov chain to eliminate sampling over the continuous state space of the targets. A major contribution of this work is the use of sparse least squares updating and downdating techniques, which significantly reduce the computational cost per iteration of the Markov chain. Also, when combined with a simple heuristic, they enable the algorithm to correctly focus computation on interacting targets. We include experimental results on a challenging simulation sequence. We test the accuracy of the algorithm using two sensor modalities, video, and laser range data. We also show the algorithm exhibits real time performance on a conventional PC},
	keywords = {algorithms, approximate inference, Artificial intelligence, auxiliary variable particle filter, Computational efficiency, continuous state space, downdating, Image Enhancement, Image Interpretation, Computer-Assisted, Inference algorithms, Information Storage and Retrieval, laser range scanner, laser range scanner., Least squares approximation, least squares approximations, Least squares methods, linear least squares, Markov chain Monte Carlo, Markov processes, MCMC data association, merged measurements, Monte Carlo methods, Movement, multiple merged measurements, multitarget tracking, particle filter, particle filtering (numerical methods), Particle filters, Pattern Recognition, Automated, probabilistic model, QR factorization, Radar tracking, Rao-Blackwellized, real time multitarget tracking, Reproducibility of results, Sampling methods, Sensitivity and Specificity, sensor fusion, sparse factorization updating, sparse least squares, State-space methods, Subtraction Technique, target tracking, updating},
	isbn = {0162-8828},
	author = {Zia Khan and Balch, T. and Dellaert, F.}
}
@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.}
}
@article {15784,
	title = {Parallel QR factorization by householder and modified Gram-Schmidt algorithms},
	journal = {Parallel Computing},
	volume = {16},
	year = {1990},
	month = {1990/11//},
	pages = {99 - 112},
	abstract = {In this paper, the parallel implementation of two algorithms for forming a QR factorization of a matrix is studied. We propose parallel algorithms for the modified Gram-Schmidt and the Householder algorithms on message passing systems in which the matrix is distributed by blocks or rows. The models that predict performance of the algorithms are validated by experimental results on several parallel machines.},
	keywords = {Gram-Schmidt algorithm, Householder algorithm, Message passing systems, QR factorization},
	isbn = {0167-8191},
	doi = {10.1016/0167-8191(90)90163-4},
	url = {http://www.sciencedirect.com/science/article/pii/0167819190901634},
	author = {O{\textquoteright}Leary, Dianne P. and Whitman,Peter}
}
