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.

Generic SoC QR array processor for adaptive beamforming

Abstract: A generic architecture for implementing a QR array processor in silicon is presented. This improves on previous research by considerably simplifying the derivation of timing schedules for a QR system implemented as a folded linear array, where account has to be taken of processor cell latency and timing at the detailed circuit level. The architecture and scheduling derived have been used to create a generator for the rapid design of System-on-a-Chip (SoC) cores for QR decomposition. This is demonstrated through the design of a single-chip architecture for implementing an adaptive beamformer for radar applications.

An Exact Least Trimmed Squares Algorithm for a Range of Coverage Values

Abstract: A new algorithm to solve exact least trimmed squares (LTS) regression is presented. The adding row algorithm (ARA) extends existing methods that compute the LTS estimator for a given coverage. It employs a tree-based strategy to compute a set of LTS regressors for a range of coverage values. Thus, prior knowledge of the optimal coverage is not required. New nodes in the regression tree are generated by updating the QR decomposition of the data matrix after adding one observation to the regression model. The ARA is enhanced by employing a branch and bound strategy. The branch and bound algorithm is an exhaustive algorithm that uses a cutting test to prune nonoptimal subtrees. It significantly improves over the ARA in computational performance. Observation preordering throughout the traversal of the regression tree is investigated. A computationally efficient and numerically stable calculation of the bounds using Givens rotations is designed around the QR decomposition, avoiding the need to explicitly updat...

QR-Decomposition-Aided Tabu Search Detection for Large MIMO Systems

Abstract: In the conventional tabu search (TS) detection algorithm for multiple-input multiple-output (MIMO) systems, the cost metrics of all neighboring vectors are computed to determine the best neighbor. This can require an excessively high computational complexity, especially in large MIMO systems because the number of neighboring vectors and the dimension per vector are large. In this study, we propose an improved TS algorithm based on the QR decomposition of the channel matrix (QR-TS), which allows for finding the best neighbor with a significantly lower complexity compared with the conventional TS algorithm. Specifically, QR-TS does not compute all metrics by early rejecting unpromising neighbors, which reduces the computational load of TS without causing any performance loss. To further optimize the QR-TS algorithm, we investigate novel ordering schemes, namely the transmit-ordering (Tx-ordering) and receive-ordering (Rx-ordering), which can considerably reduce the complexity of QR-TS. Simulation results show that QR-TS reduces the complexity approximately by a factor of two compared with the conventional TS. Furthermore, when both Tx-ordering and Rx-ordering are applied, QR-TS requires approximately $60\%\text{ -- }90\%$ less complexity compared with the conventional TS scheme. The proposed algorithms are suitable for both low-order and high-order modulation, and can achieve a significant complexity reduction compared to the Schnorr–Euchner and $K\text{-}$ best sphere decoders in large MIMO systems.

Efficient Computation of Option Prices and Greeks by Quasi--Monte Carlo Method with Smoothing and Dimension Reduction

Abstract: Discontinuities and high dimensionality are common in the problems of pricing and hedging of derivative securities. Both factors have a tremendous impact on the accuracy of the quasi--Monte Carlo (QMC) method. An ideal approach to improve the QMC method is to transform the functions to make them smoother and having smaller effective dimension. This paper develops a two-step procedure to tackle the challenging problems of both the discontinuity and the high dimensionality concurrently. In the first step, we adopt the smoothing method to remove the discontinuities of the payoff function, improving the smoothness. In the second step, we propose a general dimension reduction method (called the CQR method) to reduce the effective dimension such that the better quality of QMC points in their initial dimensions can be fully exploited. The CQR method is based on the combination of the $k$-means clustering algorithm and the QR decomposition. The $k$-means clustering algorithm, a classical algorithm of machine lear...

Stability Analysis of QR factorization in an Oblique Inner Product

Abstract: In this paper we consider the stability of the QR factorization in an oblique inner product. The oblique inner product is defined by a symmetric positive definite matrix A. We analyze two algorithm that are based a factorization of A and converting the problem to the Euclidean case. The two algorithms we consider use the Cholesky decomposition and the eigenvalue decomposition. We also analyze algorithms that are based on computing the Cholesky factor of the normal equa- tion. We present numerical experiments to show the error bounds are tight. Finally we present performance results for these algorithms as well as Gram-Schmidt methods on parallel architecture. The performance experiments demonstrate the benefit of the communication avoiding algorithms.