randUTV: A blocked randomized algorithm for computing a rank-revealing UTV factorization

This manuscript describes the randomized algorithm randUTV for computing a so called UTV factorization efficiently. Given a matrix $A$, the algorithm computes a factorization $A = UTV^{*}$, where $U$ and $V$ have orthonormal columns, and $T$ is triangular (either upper or lower, whichever is preferred). The algorithm randUTV is developed primarily to be a fast and easily parallelized alternative to algorithms for computing the Singular Value Decomposition (SVD). randUTV provides accuracy very close to that of the SVD for problems such as low-rank approximation, solving ill-conditioned linear systems, determining bases for various subspaces associated with the matrix, etc. Moreover, randUTV produces highly accurate approximations to the singular values of $A$. Unlike the SVD, the randomized algorithm proposed builds a UTV factorization in an incremental, single-stage, and non-iterative way, making it possible to halt the factorization process once a specified tolerance has been met. Numerical experiments comparing the accuracy and speed of randUTV to the SVD are presented. These experiments demonstrate that in comparison to column pivoted QR, which is another factorization that is often used as a relatively economic alternative to the SVD, randUTV compares favorably in terms of speed while providing far higher accuracy

randUTV: A Blocked Randomized Algorithm for Computing a Rank-Revealing UTV Factorization

A randomized algorithm for computing a so-called UTV factorization efficiently is presented. Given a matrix , the algorithm “randUTV” computes a factorization , where and have orthonormal columns, and is triangular (either upper or lower, whichever is preferred). The algorithm randUTV is developed primarily to be a fast and easily parallelized alternative to algorithms for computing the Singular Value Decomposition (SVD). randUTV provides accuracy very close to that of the SVD for problems such as low-rank approximation, solving ill-conditioned linear systems, and determining bases for various subspaces associated with the matrix. Moreover, randUTV produces highly accurate approximations to the singular values of . Unlike the SVD, the randomized algorithm proposed builds a UTV factorization in an incremental, single-stage, and noniterative way, making it possible to halt the factorization process once a specified tolerance has been met. Numerical experiments comparing the accuracy and speed of randUTV to the SVD are presented. Other experiments also demonstrate that in comparison to column-pivoted QR, which is another factorization that is often used as a relatively economic alternative to the SVD, randUTV compares favorably in terms of speed while providing far higher accuracy

A Pipeline for the QR Update in Digital Signal Processing

[EN] The input and output signals of a digital signal processing system can often be represented by a rectangular matrix as it is the case of the beamformer algorithm, a very useful particular algorithm that allows extraction of the original input signal once it is cleaned from noise and room reverberation. We use a version of this algorithm in which the system matrix must be factorized to solve a least squares problem. The matrix changes periodically according to the input signal sampled; therefore, the factorization needs to be recalculated as fast as possible. In this paper, we propose to use parallelism through a pipeline pattern. With our pipeline, some partial computations are advanced so that the final time required to update the factorization is highly reducedThis work was supported by the Spanish Ministry of Economy and Competitiveness under MINECO and FEDER projects TIN2014-53495-R and TEC2015-67387-C4-1-R.Dolz, MF.; Alventosa, FJ.; Alonso-Jordá, P.; Vidal Maciá, AM. (2019). A Pipeline for the QR Update in Digital Signal Processing. Computational and Mathematical Methods. 1:1-13. https://doi.org/10.1002/cmm4.1022S113

Randomized methods for matrix computations

The purpose of this text is to provide an accessible introduction to a set of recently developed algorithms for factorizing matrices. These new algorithms attain high practical speed by reducing the dimensionality of intermediate computations using randomized projections. The algorithms are particularly powerful for computing low-rank approximations to very large matrices, but they can also be used to accelerate algorithms for computing full factorizations of matrices. A key competitive advantage of the algorithms described is that they require less communication than traditional deterministic methods