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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.


Papers
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Journal ArticleDOI
TL;DR: From this point of view it is seen that the lattice algorithm is really an efficient way of solving specially structured least-squares problems by orthogonalization as opposed to solving the normal equations by fast Toeplitz algorithms.
Abstract: A new orthogonalization technique is presented for computing the QR factorization of a general n X p matrix of full rank p (n 2 p). The method is based on the use of projections to solve increasingly larger subproblems recursively and has an O(np2) operation count for general matrices. The technique is readily adaptable to solving linear least-squares problems. If the initial matrix has a circulant structure the algorithm simplifies significantly and gives the so-called lattice algorithm for solving linear prediction problems. From this point of view it is seen that the lattice algorithm is really an efficient way of solving specially structured least-squares problems by orthogonalization as opposed to solving the normal equations by fast Toeplitz algorithms.

64 citations

Journal ArticleDOI
TL;DR: In this paper, first-order perturbation analysis for QR matrix factorization is presented for a given real matrix of rank n and general perturbations in rank n which are sufficiently small in norm.
Abstract: This paper gives perturbation analyses for $Q_1$ and $R$ in the QR factorization $A=Q_1R$, $Q_1^TQ_1=I$ for a given real $m\times n$ matrix $A$ of rank $n$ and general perturbations in $A$ which are sufficiently small in norm. The analyses more accurately reflect the sensitivity of the problem than previous such results. The condition numbers here are altered by any column pivoting used in $AP=Q_1R$, and 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 $Q_1$, so the computed $Q_1$ and $R$ will probably both have greatest accuracy when we use the standard column pivoting strategy. First-order perturbation analyses are given for both $Q_1$ and $R$. It is seen that the analysis for $R$ may be approached in two ways---a detailed "matrix--vector equation" analysis which provides a tight bound and corresponding condition number, which unfortunately is costly to compute and not very intuitive, and a simpler "matrix equation" analysis which provides results that are usually weaker but easier to interpret and which allows the efficient computation of satisfactory estimates for the actual condition number. These approaches are powerful general tools and appear to be applicable to the perturbation analysis of any matrix factorization.

64 citations

Book
01 Jan 2009
TL;DR: QR Decomposition An Annotated Bibliography contains references to Adaptive Filters, Conventional and Inverse QRD-RLS Algorithms, and Numerical Stability Properties.
Abstract: QR Decomposition An Annotated Bibliography.- to Adaptive Filters.- Conventional and Inverse QRD-RLS Algorithms.- Fast QRD-RLS Algorithms.- QRD Least-Squares Lattice Algorithms.- Multichannel Fast QRD-RLS Algorithms.- Householder-Based RLS Algorithms.- Numerical Stability Properties.- Finite and Infinite-Precision Properties of QRD-RLS Algorithms.- On Pipelined Implementations of QRD-RLS Adaptive Filters.- Weight Extraction of Fast QRD-RLS Algorithms.- Linear Constrained QRD-Based Algorithm.

63 citations

Journal ArticleDOI
TL;DR: The Halley iteration can be implemented via QR decompositions without explicit matrix inversions, and it is an inverse free communication friendly algorithm for the emerging multicore and hybrid high performance computing systems.
Abstract: We introduce a dynamically weighted Halley (DWH) iteration for computing the polar decomposition of a matrix, and we prove that the new method is globally and asymptotically cubically convergent. For matrices with condition number no greater than $10^{16}$, the DWH method needs at most six iterations for convergence with the tolerance $10^{-16}$. The Halley iteration can be implemented via QR decompositions without explicit matrix inversions. Therefore, it is an inverse free communication friendly algorithm for the emerging multicore and hybrid high performance computing systems.

63 citations

Journal ArticleDOI
TL;DR: It is shown that by adding a correction step using only single precision the authors get a method which under mild conditions is as accurate as the QR method.

63 citations


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