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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
01 Apr 2020
TL;DR: The LU-Cholesky QR algorithms for thin QR decomposition are proposed, i.e., the idea is to use LU-factors of a given matrix as preconditioning before applying Cholesky decomposition.
Abstract: This paper aims to propose the LU-Cholesky QR algorithms for thin QR decomposition (also called economy size or reduced QR decomposition). CholeskyQR is known as a fast algorithm employed for thin QR decomposition, and CholeskyQR2 aims to improve the orthogonality of a Q-factor computed by CholeskyQR. Although such Cholesky QR algorithms can efficiently be implemented in high-performance computing environments, they are not applicable for ill-conditioned matrices, as compared to the Householder QR and the Gram–Schmidt algorithms. To address this problem, we apply the concept of LU decomposition to the Cholesky QR algorithms, i.e., the idea is to use LU-factors of a given matrix as preconditioning before applying Cholesky decomposition. Moreover, we present rounding error analysis of the proposed algorithms on the orthogonality and residual of computed QR-factors. Numerical examples provided in this paper illustrate the efficiency of the proposed algorithms in parallel computing on both shared and distributed memory computers.

12 citations

Journal ArticleDOI
TL;DR: Two new algorithms for computing the 1-norm condition number in O(n) operations are devised, which avoid underflow and overflow, and are simpler than existing algorithms since tests are not required for degenerate cases.
Abstract: For an $n \times n$ tridiagonal matrix we exploit the structure of its QR factorization to devise two new algorithms for computing the 1-norm condition number in $O(n)$ operations. The algorithms avoid underflow and overflow, and are simpler than existing algorithms since tests are not required for degenerate cases. An error analysis of the first algorithm is given, while the second algorithm is shown to be competitive in speed with existing algorithms. We then turn our attention to an $n \times n$ diagonal-plus-semiseparable matrix, $A$, for which several algorithms have recently been developed to solve $Ax=b$ in $O(n)$ operations. We again exploit the QR factorization of the matrix to present an algorithm that computes the 1-norm condition number in $O(n)$ operations.

12 citations

Journal ArticleDOI
TL;DR: The algorithms, which are extensions of the rank-1 updating method, achieve the updating using approximately 2(k + n) compound disjoint Givens rotations (CDGRs) with elements annihilated by rotations in adjacent planes.
Abstract: Parallel strategies based on Givens rotations are proposed for updating the QR decomposition of an n × n matrix after a rank-k change (k < n). The complexity analyses of the Givens algorithms are based on the total number of Givens rotations applied to a 2-element vector. The algorithms, which are extensions of the rank-1 updating method, achieve the updating using approximately 2(k + n) compound disjoint Givens rotations (CDGRs) with elements annihilated by rotations in adjacent planes. Block generalization of the serial rank-1 algorithms are also presented. The algorithms are rich in level 3 BLAS operations, making them suitable for implementation on large scale parallel systems. The performance of some of the algorithms on a 2-D SIMD (single instruction stream--multiple instruction stream) array processor is discussed.

12 citations

Posted Content
TL;DR: In this paper, an exponential analysis method to retrieve high-resolution information from coarse-scale measurements, using uniform downsampling, is presented, which can be combined with different existing implementations of multiexponential analysis (matrix pencil, MUSIC, ESPRIT, APM, generalized overdetermined eigenvalue solver, simultaneous QR factorization,...) and so is very versatile.
Abstract: Sampling a signal below the Shannon-Nyquist rate causes aliasing, meaning different frequencies to become indistinguishable. It is also well-known that recovering spectral information from a signal using a parametric method can be ill-posed or ill-conditioned and therefore should be done with caution. We present an exponential analysis method to retrieve high-resolution information from coarse-scale measurements, using uniform downsampling. We exploit rather than avoid aliasing. While we loose the unicity of the solution by the downsampling, it allows to recondition the problem statement and increase the resolution. Our technique can be combined with different existing implementations of multi-exponential analysis (matrix pencil, MUSIC, ESPRIT, APM, generalized overdetermined eigenvalue solver, simultaneous QR factorization,...) and so is very versatile. It seems to be especially useful in the presence of clusters of frequencies that are difficult to distinguish from one another.

12 citations

Journal ArticleDOI
TL;DR: Simulation results show that both of the proposed decoders enable the QSTBC to achieve ML performance with significant reduction in computational load.
Abstract: This letter proposes two very-low-complexity maximum-likelihood (ML) detection algorithms based on QR decomposition for the quasi-orthogonal space-time code (QSTBC) with four transmit antennas [3]-[5], called VLCMLDec1 and VLCMLDec2 decoders. The first decoder, VLCMLDecl, can be used to detect transmitted symbols being extracted from finite-size constellations such as phase-shift keying (PSK) or quadrature amplitude modulation (QAM). The second decoder, VL-CMLDec2, is an enhanced version of the VLCMLDecl, developed mainly for QAM constellations. Simulation results show that both of the proposed decoders enable the QSTBC to achieve ML performance with significant reduction in computational load.

12 citations


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