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.

Parallel out-of-core cholesky and QR factorizations with POOCLAPACK

Abstract: In this paper the parallel implementation of out-of-core Cholesky factorization is used to introduce the Parallel Outof-Core Linear Algebra Package (POOCLAPACK), a flexible infrastructure for parallel implementation of out-ofcore linear algebra operations. POOCLAPACK builds on the Parallel Linear Algebra Package (PLAPACK) for incore parallel dense linear algebra computation. Despite the extreme simplicity of POOCLAPACK, the out-of-core Cholesky factorization implementation is shown to achieve up to 80% of peak performance on a 64 node configuration of the Cray T3E-600. The insights gained from examining the Cholesky factorization are also applied to the much more difficult and important QR factorization operation. Preliminary results for parallel implementation of the resulting OOC QR factorization algorithm are included.

A New Adaptive Robust Unscented Kalman Filter for Improving the Accuracy of Target Tracking

Abstract: In target tracking, the tracking process needs to constantly update the data information. However, during data acquisition and transmission of sensors, outliers may occur frequently, and the model is disturbed by non-Gaussian noise, that affects the performance of system state estimation. In this paper, a new filtering algorithm is proposed based on QR decomposition and singular value decomposition (SVD) method, namely adaptive robust unscented Kalman filter (QS-ARUKF) to suppress the interference of outliers, nonGaussian noise as well as a model error to achieve high accuracy state estimation. An adaptive filtering algorithm based on strong tracking idea is used in modifying the state equation of unscented Kalman filter (UKF), so that the algorithm can effectively improve the tracking ability of the state model. By using the robust filtering method to construct a new cost function used to modify the measurement covariance formula of the Kalman filter, the error of measurement model can be effectively suppressed. The QR decomposition is introduced to the time update and measurement update to avoid the covariance non-positive definite. We propose the SVD method to address the problem of numerical sensitivity in the filtering process. The purpose of this method is to replace the calculation of the inverse of the filter gain matrix and further improve the robustness of the algorithm. The simulation results showed that the proposed algorithm has higher accuracy and better robustness than the traditional filtering method.

Nonlinear QR code based optical image encryption using spiral phase transform, equal modulus decomposition and singular value decomposition

Abstract: In this study, we propose a quick response (QR) code based nonlinear optical image encryption technique using spiral phase transform (SPT), equal modulus decomposition (EMD) and singular value decomposition (SVD). First, the primary image is converted into a QR code and then multiplied with a spiral phase mask (SPM). Next, the product is spiral phase transformed with particular spiral phase function, and further, the EMD is performed on the output of SPT, which results into two complex images, Z 1 and Z 2. Among these, Z 1 is further Fresnel propagated with distance d, and Z 2 is reserved as a decryption key. Afterwards, SVD is performed on Fresnel propagated output to get three decomposed matrices i.e. one diagonal matrix and two unitary matrices. The two unitary matrices are modulated with two different SPMs and then, the inverse SVD is performed using the diagonal matrix and modulated unitary matrices to get the final encrypted image. Numerical simulation results confirm the validity and effectiveness of the proposed technique. The proposed technique is robust against noise attack, specific attack, and brutal force attack. Simulation results are presented in support of the proposed idea.

Blocked rank-revealing QR factorizations: How randomized sampling can be used to avoid single-vector pivoting

Abstract: Given a matrix $A$ of size $m\times n$, the manuscript describes a algorithm for computing a QR factorization $AP=QR$ where $P$ is a permutation matrix, $Q$ is orthonormal, and $R$ is upper triangular. The algorithm is blocked, to allow it to be implemented efficiently. The need for single vector pivoting in classical algorithms for computing QR factorizations is avoided by the use of randomized sampling to find blocks of pivot vectors at once. The advantage of blocking becomes particularly pronounced when $A$ is very large, and possibly stored out-of-core, or on a distributed memory machine. The manuscript also describes a generalization of the QR factorization that allows $P$ to be a general orthonormal matrix. In this setting, one can at moderate cost compute a \textit{rank-revealing} factorization where the mass of $R$ is concentrated to the diagonal entries. Moreover, the diagonal entries of $R$ closely approximate the singular values of $A$. The algorithms described have asymptotic flop count $O(m\,n\,\min(m,n))$, just like classical deterministic methods. The scaling constant is slightly higher than those of classical techniques, but this is more than made up for by reduced communication and the ability to block the computation.