Probabilistic Analysis of Gaussian Elimination Without Pivoting
SIAM Journal on Matrix Analysis and Applications
Statistical Condition Estimation for Linear Least Squares
SIAM Journal on Matrix Analysis and Applications
LAPACK Users' guide (third ed.)
LAPACK Users' guide (third ed.)
Accuracy and Stability of Numerical Algorithms
Accuracy and Stability of Numerical Algorithms
Iterative Methods for Sparse Linear Systems
Iterative Methods for Sparse Linear Systems
Monte Carlo Methods for Applied Scientists
Monte Carlo Methods for Applied Scientists
A Partial Condition Number for Linear Least Squares Problems
SIAM Journal on Matrix Analysis and Applications
Communication avoiding Gaussian elimination
Proceedings of the 2008 ACM/IEEE conference on Supercomputing
Programming matrix algorithms-by-blocks for thread-level parallelism
ACM Transactions on Mathematical Software (TOMS)
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We illustrate how linear algebra calculations can be enhanced by statistical techniques in the case of a square linear system Ax = b. We study a random transformation of A that enables us to avoid pivoting and then to reduce the amount of communication. Numerical experiments show that this randomization can be performed at a very affordable computational price while providing us with a satisfying accuracy when compared to partial pivoting. This random transformation called Partial Random Butterfly Transformation (PRBT) is optimized in terms of data storage and flops count. We propose a solver where PRBT and the LU factorization with no pivoting take advantage of the current hybrid multicore/GPU machines and we compare its Gflop/s performance with a solver implemented in a current parallel library.