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.

Distributed QR factorization based on randomized algorithms

Abstract: Most parallel algorithms for matrix computations assume a static network with reliable communication and thus use fixed communication schedules. However, in situations where computer systems may change dynamically, in particular, when they have unreliable components, algorithms with randomized communication schedule may be an interesting alternative. We investigate randomized algorithms based on gossiping for the distributed computation of the QR factorization. The analyses of numerical accuracy showed that known numerical properties of classical sequential and parallel QR decomposition algorithms are preserved. Moreover, we illustrate that the randomized approaches are well suited for distributed systems with arbitrary topology and potentially unreliable communication, where approaches with fixed communication schedules have major drawbacks. The communication overhead compared to the optimal parallel QR decomposition algorithm (CAQR) is analyzed. The randomized algorithms have a much higher potential for trading off numerical accuracy against performance because their accuracy is proportional to the amount of communication invested.

Parallel tiled QR factorization for multicore architectures

Abstract: As multicore systems continue to gain ground in the high-performance computing world, linear algebra algorithms have to be reformulated or new algorithms have to be developed in order to take advan...

Robust partial pole assignment for vibrating systems with aerodynamic effects

Abstract: This paper proposes a novel algorithm for robust partial eigenvalue assignment (RPEVA) problem for a cubic matrix pencil arising from modeling of vibrating systems with aerodynamic effects. The RPEVA problem for a cubic pencil is the one of choosing suitable feedback matrices to reassign a few (say k < 3n) unwanted eigenvalues while leaving the remaining large number (3n - k) of them unchanged, in such a way that the the eigenvalues of the closed-loop matrix are as insensitive as possible to small perturbation of the data. The latter amounts to minimizing the condition number of the closed-loop eigenvector matrix. The problem is solved directly in the cubic matrix polynomial setting without making any transformation to a standard first-order state-space system. This allows us to take advantage of the exploitable structures such as the sparsity, definiteness, bandness, etc., very often offered by large practical problems. The major computational requirements are: (i)solutions of a small Sylvester equation, (ii) QR factorizations, and (iii) solutions or some standard least squares problems. The least-squares problems result from matrix rank-one and rank-two update techniques used in the algorithm for reassigning, respectively, simple and complex eigenvalues. The practical effectiveness of the method is demonstrated by implementational results on simulated data provided by the Boeing company.

Simple, efficient and robust techniques for automatic multi-objective function parameterisation: case studies of local and global optimisation using APSIM

Abstract: Several techniques for automatic parameterisation are explored using the software PEST. We parameterised the biophysical systems model APSIM with measurements from a maize cropping experiment with the objective of finding algorithms that resulted in the least distance between modelled and measured data (φ) in the shortest possible time. APSIM parameters were optimised using a weighted least-squares approach that minimised the value of φ. Optimisation techniques included the Gauss-Marquardt-Levenberg (GML) algorithm, singular value decomposition (SVD), least squares with QR decomposition (LSQR), Tikhonov regularisation, and covariance matrix adaptation-evolution strategy (CMAES). In general, CMAES with log transformed APSIM parameters and larger population size resulted in the lowest φ, but this approach required significantly longer to converge compared with other optimisation algorithms. Regularisation treatments with log transformed parameters also resulted in low φ values when combined with SVD or LSQR; LSQR treatments with no regularisation tended to converge earliest. In addition to an analysis of several PEST algorithms, this study provides a narrative on how methodologies presented here could be generalised and applied to other models.

Analysis of a QR Algorithm for Computing Singular Values

Abstract: We extend the Golub--Kahan algorithm for computing the singular value decomposition of bidiagonal matrices to triangular matrices $R$. Our algorithm avoids the explicit formation of $R^TR$ or $RR^T$. We derive a relation between left and right singular vectors of triangular matrices and use it to prove monotonic convergence of singular values and singular vectors. The convergence rate for singular values equals the square of the convergence rate for singular vectors. The convergence behaviour explains the occurrence of deflation in the interior of the matrix. We analyse the relationship between our algorithm and rank-revealing QR and URV decompositions. As a consequence, we obtain an algorithm for computing the URV decomposition, as well as a divide-and-conquer algorithm that computes singular values of dense matrices and may be beneficial on a parallel architecture. Our perturbation result for the smallest singular values of a triangular matrix is stronger than the traditional results because it guarantees high relative accuracy in the smallest singular values after an off-diagonal block of the matrix has been set to zero.