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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.


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Journal ArticleDOI
TL;DR: In this paper, componentwise perturbation analyses for Q and R in the QR factorization A=QR, ��$Q^\mathrm{T}Q=I$¯¯¯¯, R upper triangular, for a given real $m\times n$ matrix A of rank n.
Abstract: This paper gives componentwise perturbation analyses for Q and R in the QR factorization A=QR, $Q^\mathrm{T}Q=I$ , R upper triangular, for a given real $m\times n$ matrix A of rank n. Such specific analyses are important for example when the columns of A are badly scaled. First order perturbation bounds are given for both Q and R. The analyses more accurately reflect the sensitivity of the problem than previous such results. The condition number for R is bounded for a fixed n when the standard column pivoting strategy is used. This strategy also tends to improve the condition of Q, so usually the computed Q and R will both have higher accuracy when we use the standard column pivoting strategy. Practical condition estimators are derived. The assumptions on the form of the perturbation $\Delta A$ are explained and extended. Weaker rigorous bounds are also given.

21 citations

Journal ArticleDOI
TL;DR: A novel technique for direction-of-arrival estimation based on computing a permutation matrix E and a QR factorization RE=HB of the permuted covariance matrix R, such that a possible rank deficiency of R is revealed in the triangular factor B having a minimum norm lower right block.
Abstract: The authors describe a novel technique for direction-of-arrival estimation based on computing a permutation matrix E and a QR factorization RE=HB of the permuted covariance matrix R, such that a possible rank deficiency of R is revealed in the triangular factor B having a minimum norm lower right block. A subset of the columns of the orthogonal matrix, H, is shown to be orthogonal to the direction vectors of sources and hence can be used to estimate their bearings. The cost of this algorithm is only slightly more than that of one QR factorization, but is much lower than that of an eigen-decomposition. Simulation results are included to show that the proposed method performs nearly as well as MUSIC in terms of signal resolution, bias, and variance of the estimated bearings. >

21 citations

Journal ArticleDOI
TL;DR: It is proved, under assumptions similar to assumptions used by others, that if the numerical rank is chosen at a gap in the singular value spectrum and if the initial factorization is rank-revealing, then, even if the algorithm is stopped after the first step, approximately half the time its solutions are closer to the desired solution than are the singularvalue decomposition (SVD) solutions.
Abstract: The algorithm of Mathias and Stewart [Linear Algebra Appl., 182 (1993), pp. 91--100] is examined as a tool for constructing regularized solutions to rank-deficient and ill-posed linear equations. The algorithm is based on a sequence of QR factorizations. If it is stopped after the first step, it produces the same solution as the complete orthogonal decomposition used in LAPACK's xGELSY. However, we show that for low-rank problems a careful implementation can lead to an order of magnitude improvement in speed over xGELSY as implemented in LAPACK. We prove, under assumptions similar to assumptions used by others, that if the numerical rank is chosen at a gap in the singular value spectrum and if the initial factorization is rank-revealing, then, even if the algorithm is stopped after the first step, approximately half the time its solutions are closer to the desired solution than are the singular value decomposition (SVD) solutions. Conversely, the SVD will be closer approximately half the time, and in this case overall the two algorithms are very similar in accuracy. We confirm this with numerical experiments. Although the algorithm works best for problems with a gap in the singular value spectrum, numerical experiments suggest that it may work well for problems with no gap.

21 citations

01 Jan 2012
TL;DR: This paper considers computational complexity and efficacious of algorithm present a PCA/range(Sb) algorithm for dimensionality reduction of data, which transforms firstly the original space by using a basis of range(S b) and then in the transformed space applies PCA.
Abstract: Inspired and motivated by the idea of LDA/QR presented by Ye and Li, in addition, by the idea of WK- DA/QR and WKDA/SVD presented by Gao and Fan. In this paper, we first consider computational complexity and efficacious of algorithm present a PCA/range(Sb) algorithm for dimensionality reduction of data, which transforms firstly the original space by using a basis of range(Sb) and then in the transformed space applies PCA. Considering computationally expensive and time complexity, we further present an improved version of PCA/range(Sb), denot- ed by PCA/range(Sb)-QR, in which QR decomposition is used at the last step of PCA/range(Sb). In addition, we also improve LDA/GSVD, LDA/range(Sb) and PCA by means of QR decomposition. Extensive experiments on face images from UCI data sets show the effectiveness of the proposed algorithms.

21 citations

Journal ArticleDOI
TL;DR: An adaptive selection algorithm for the surviving symbol replica candidates (ASESS) based on the maximum reliability in maximum likelihood detection with QR decomposition and the M-algorithm and the QRM-MLD for Orthogonal Frequency Division Multiplexing (OFDM) multiple-input multiple-output (MIMO) multiplexing is proposed.
Abstract: This paper proposes an adaptive selection algorithm for the surviving symbol replica candidates (ASESS) based on the maximum reliability in maximum likelihood detection with QR decomposition and the M-algorithm (QRM-MLD) for Orthogonal Frequency Division Multiplexing (OFDM) multiple-input multiple-output (MIMO) multiplexing. In the proposed algorithm, symbol replica candidates newly-added at each stage are ranked for each surviving symbol replica from the previous stage using multiple quadrant detection. Then, branch metrics are calculated only for the minimum number of symbol replica candidates with a high level of reliability using an iterative loop based on symbol ranking results. Computer simulation results show that the computational complexity of the QRM-MLD employing the proposed ASESS algorithm is reduced to approximately 1/4 and 1/1200 compared to that of the original QRM-MLD and that of the conventional MLD with squared Euclidian distance calculations for all symbol replica candidates, respectively, assuming the identical achievable average packet error rate (PER) performance in 4-by-4 MIMO multiplexing with 16QAM data modulation. The results also show that 1-Gbps throughput is achieved at the average received signal energy per bit-to-noise power spectrum density ratio (Eb/N0) per receiver antenna of approximately 9dB using the ASESS algorithm in QRM-MLD associated with 16QAM modulation and Turbo coding with the coding rate of 8/9 assuming a 100-MHz bandwidth for a 12-path Rayleigh fading channel (root mean square (r.m.s.) delay spread of 0.26µs and maximum Doppler frequency of 20Hz).

21 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
202331
202273
202190
2020132
2019126
2018139