QR decomposition

About: QR decomposition is a research topic. Over the lifetime, 3504 publications have been published within this topic receiving 100599 citations. The topic is also known as: QR factorization.

Blendenpik: Supercharging LAPACK's Least-Squares Solver

Abstract: Several innovative random-sampling and random-mixing techniques for solving problems in linear algebra have been proposed in the last decade, but they have not yet made a significant impact on numerical linear algebra. We show that by using a high-quality implementation of one of these techniques, we obtain a solver that performs extremely well in the traditional yardsticks of numerical linear algebra: it is significantly faster than high-performance implementations of existing state-of-the-art algorithms, and it is numerically backward stable. More specifically, we describe a least-squares solver for dense highly overdetermined systems that achieves residuals similar to those of direct QR factorization-based solvers (lapack), outperforms lapack by large factors, and scales significantly better than any QR-based solver.

Some applications of the rank revealing QR factorization

Abstract: The rank revealing QR factorization of a rectangular matrix can sometimes be used as a reliable and efficient computational alternative to the singular value decomposition for problems that involve rank determination. This is illustrated by showing how the rank revealing QR factorization can be used to compute solutions to rank deficient least squares problems, to perform subset selection, to compute matrix approximations of given rank, and to solve total least squares problems.

Algorithm-based fault detection for signal processing applications

Abstract: The increasing demands for high-performance signal processing along with the availability of inexpensive high-performance processors have results in numerous proposals for special-purpose array processors for signal processing applications. A functional-level concurrent error-detection scheme is presented for such VLSI signal processing architectures as those proposed for the FFT and QR factorization. Some basic properties involved in such computations are used to check the correctness of the computed output values. This fault-detection scheme is shown to be applicable to a class of problems rather than a particular problem, unlike the earlier algorithm-based error-detection techniques. The effects of roundoff/truncation errors due to finite-precision arithmetic are evaluated. It is shown that the error coverage is high with large word sizes. >

A New Selection Operator for the Discrete Empirical Interpolation Method -- improved a priori error bound and extensions

Abstract: This paper introduces a new framework for constructing the Discrete Empirical Interpolation Method DEIM projection operator. The interpolation node selection procedure is formulated using the QR factorization with column pivoting, and it enjoys a sharper error bound for the DEIM projection error. Furthermore, for a subspace $\mathcal{U}$ given as the range of an orthonormal $U$, the DEIM projection does not change if $U$ is replaced by $U \Omega$ with arbitrary unitary matrix $\Omega$. In a large-scale setting, the new approach allows modifications that use only randomly sampled rows of $U$, but with the potential of producing good approximations with corresponding probabilistic error bounds. Another salient feature of the new framework is that robust and efficient software implementation is easily developed, based on readily available high performance linear algebra packages.

A survey of condition number estimation for triangular matrices

Abstract: We survey and compare a wide variety of techniques for estimating the condition number of a triangular matrix, and make recommendations concerning the use of the estimates in applications. Each of the methods is shown to bound the condition number; the bounds can broadly be categorised as upper bounds from matrix theory and lower bounds from heuristic or probabilistic algorithms. For each bound we examine by how much, at worst, it can overestimate or underestimate the condition number. Numerical experiments are presented in order to illustrate and compare the practical performance of the condition estimators.