@article {917,
	title = {Mixing LU-QR Factorization Algorithms to Design High-Performance Dense Linear Algebra Solvers},
	journal = {Journal of Parallel and Distributed Computing},
	volume = {85},
	year = {2015},
	month = {2015-11},
	pages = {32-46},
	abstract = {This paper introduces hybrid LU{\textendash}QR algorithms for solving dense linear systems of the form Ax=b. Throughout a matrix factorization, these algorithms dynamically alternate LU with local pivoting and QR elimination steps based upon some robustness criterion. LU elimination steps can be very efficiently parallelized, and are twice as cheap in terms of floating-point operations, as QR steps. However, LU steps are not necessarily stable, while QR steps are always stable. The hybrid algorithms execute a QR step when a robustness criterion detects some risk for instability, and they execute an LU step otherwise. The choice between LU and QR steps must have a small computational overhead and must provide a satisfactory level of stability with as few QR steps as possible. In this paper, we introduce several robustness criteria and we establish upper bounds on the growth factor of the norm of the updated matrix incurred by each of these criteria. In addition, we describe the implementation of the hybrid algorithms through an extension of the PaRSEC software to allow for dynamic choices during execution. Finally, we analyze both stability and performance results compared to state-of-the-art linear solvers on parallel distributed multicore platforms. A comprehensive set of experiments shows that hybrid LU{\textendash}QR algorithms provide a continuous range of trade-offs between stability and performances.},
	keywords = {lu factorization, Numerical algorithms, QR factorization, Stability; Performance},
	doi = {doi:10.1016/j.jpdc.2015.06.007},
	author = {Mathieu Faverge and Julien Herrmann and Julien Langou and Bradley Lowery and Yves Robert and Jack Dongarra}
}
@conference {813,
	title = {Designing LU-QR Hybrid Solvers for Performance and Stability},
	booktitle = {IPDPS 2014},
	year = {2014},
	month = {2014-05},
	publisher = {IEEE},
	organization = {IEEE},
	address = {Phoenix, AZ},
	abstract = {This paper introduces hybrid LU-QR algorithms for solving dense linear systems of the form Ax = b. Throughout a matrix factorization, these algorithms dynamically alternate LU with local pivoting and QR elimination steps, based upon some robustness criterion. LU elimination steps can be very efficiently parallelized, and are twice as cheap in terms of operations, as QR steps. However, LU steps are not necessarily stable, while QR steps are always stable. The hybrid algorithms execute a QR step when a robustness criterion detects some risk for instability, and they execute an LU step otherwise. Ideally, the choice between LU and QR steps must have a small computational overhead and must provide a satisfactory level of stability with as few QR steps as possible. In this paper, we introduce several robustness criteria and we establish upper bounds on the growth factor of the norm of the updated matrix incurred by each of these criteria. In addition, we describe the implementation of the hybrid algorithms through an extension of the Parsec software to allow for dynamic choices during execution. Finally, we analyze both stability and performance results compared to state-of-the-art linear solvers on parallel distributed multicore platforms.},
	keywords = {plasma},
	isbn = {978-1-4799-3800-1},
	doi = {10.1109/IPDPS.2014.108},
	author = {Mathieu Faverge and Julien Herrmann and Julien Langou and Bradley Lowery and Yves Robert and Jack Dongarra}
}
@techreport {703,
	title = {Designing LU-QR hybrid solvers for performance and stability},
	journal = {University of Tennessee Computer Science Technical Report (also LAWN 282)},
	number = {ut-eecs-13-719},
	year = {2013},
	month = {2013-10},
	publisher = {University of Tennessee},
	author = {Mathieu Faverge and Julien Herrmann and Julien Langou and Bradley Lowery and Yves Robert and Jack Dongarra}
}
