Direct QR factorizations for tall-and-skinny matrices in MapReduce architectures

The QR factorization and the SVD are two fundamental matrix decompositions with applications throughout scientific computing and data analysis. For matrices with many more rows than columns, so-called "tall-and-skinny matrices," there is a numerically stable, efficient, communication-avoiding algorithm for computing the QR factorization. It has been used in traditional high performance computing and grid computing environments. For MapReduce environments, existing methods to compute the QR decomposition use a numerically unstable approach that relies on indirectly computing the Q factor. In the best case, these methods require only two passes over the data. In this paper, we describe how to compute a stable tall-and-skinny QR factorization on a MapReduce architecture in only slightly more than 2 passes over the data. We can compute the SVD with only a small change and no difference in performance. We present a performance comparison between our new direct TSQR method, a standard unstable implementation for MapReduce (Cholesky QR), and the classic stable algorithm implemented for MapReduce (Householder QR). We find that our new stable method has a large performance advantage over the Householder QR method. This holds both in a theoretical performance model as well as in an actual implementation

A weakly stable algorithm for general Toeplitz systems

We show that a fast algorithm for the QR factorization of a Toeplitz or Hankel matrix A is weakly stable in the sense that R^T.R is close to A^T.A. Thus, when the algorithm is used to solve the semi-normal equations R^T.Rx = A^Tb, we obtain a weakly stable method for the solution of a nonsingular Toeplitz or Hankel linear system Ax = b. The algorithm also applies to the solution of the full-rank Toeplitz or Hankel least squares problem.Comment: 17 pages. An old Technical Report with postscript added. For further details, see http://wwwmaths.anu.edu.au/~brent/pub/pub143.htm

Tall-and-skinny QR factorization with approximate Householder reflectors on graphics processors

[EN] We present a novel method for the QR factorization of large tall-and-skinny matrices that introduces an approximation technique for computing the Householder vectors. This approach is very competitive on a hybrid platform equipped with a graphics processor, with a performance advantage over the conventional factorization due to the reduced amount of data transfers between the graphics accelerator and the main memory of the host. Our experiments show that, for tall¿skinny matrices, the new approach outperforms the code in MAGMA by a large margin, while it is very competitive for square matrices when the memory transfers and CPU computations are the bottleneck of the Householder QR factorizationThis research was supported by the Project TIN2017-82972-R from the MINECO (Spain) and the EU H2020 Project 732631 "OPRECOMP. Open Transprecision Computing".Tomás Domínguez, AE.; Quintana-Ortí, ES. (2020). 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