Topic
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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01 Jan 1982
TL;DR: A concurrent algorithm and corresponding array for computing the triangular matrix R by Householder transformations is described and particular attention is given to issues such as broadcasting and pipelining.
Abstract: The QR-method is a method for the solution of linear system of equations. The matrix R is upper triangular and Q is a unitary matrix. In equation solving Q is not always computed explicitly. The matrix R can be obtained by applying a sequence of unitary transformations to the matrix defining the system of equations. Householder's method or Given's method can be used to determine
unitary transformation matrices. This paper describes a concurrent algorithm and corresponding array for computing the triangular matrix R by Householder transformations. Particular attention is given to issues such as broadcasting
and pipelining.
31 citations
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TL;DR: An in-depth complexity analysis shows that the proposed algorithms, for a sufficiently large number of data-carrying tones and sufficiently small channel order, provably exhibit significantly smaller complexity than brute-force per-tone QR decomposition.
Abstract: Detection algorithms for multiple-input multiple-output (MIMO) wireless systems based on orthogonal frequency-division multiplexing (OFDM) typically require the computation of a QR decomposition for each of the data-carrying OFDM tones. The resulting computational complexity will, in general, be significant. Motivated by the fact that the channel matrices arising in MIMO-OFDM systems result from oversampling of a polynomial matrix, we formulate interpolation-based QR decomposition algorithms. An in-depth complexity analysis, based on a metric relevant for very large scale integration (VLSI) implementations, shows that the proposed algorithms, for a sufficiently large number of data-carrying tones and sufficiently small channel order, provably exhibit significantly smaller complexity than brute-force per-tone QR decomposition.
31 citations
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TL;DR: The Fast Least Squares algorithms based on the QR triangular decomposition of the input signal matrix and developed for the case of one-dimensional signals can be extended to handle the cases of multi-dimensional (MD) or multichannel signals.
31 citations
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01 Feb 2006TL;DR: It is shown experimentally, that the QR factorization with the complete column pivoting, optionally followed by the LQ factorization of the R-factor, can lead to a substantial decrease of the number of outer parallel iteration steps.
Abstract: One way, how to speed up the computation of the singular value decomposition of a given matrix [email protected]?C^m^x^n,m>=n, by the parallel two-sided block-Jacobi method, consists of applying some pre-processing steps that would concentrate the Frobenius norm near the diagonal. Such a concentration should hopefully lead to fewer outer parallel iteration steps needed for the convergence of the entire algorithm. It is shown experimentally, that the QR factorization with the complete column pivoting, optionally followed by the LQ factorization of the R-factor, can lead to a substantial decrease of the number of outer parallel iteration steps, whereby the details depend on the condition number and on the distribution of singular values including their multiplicity. A subset of ill-conditioned matrices has been identified, for which the dynamic ordering becomes inefficient. Best results in numerical experiments performed on the cluster of personal computers were achieved for well-conditioned matrices with a multiple minimal singular value, where the number of parallel iteration steps was reduced by two orders of magnitude. However, the gain in speed, as measured by the total parallel execution time, depends decisively on the implementation of the distributed QR and LQ factorizations on a given parallel architecture. In general, the reduction of the total parallel execution time up to one order of magnitude has been achieved.
31 citations
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TL;DR: The method proposed in this paper attacks such problems from the dual side, alternatively arranging computations of the simplex method using the QR factorization and a new crash heuristic, having a clear geometrical meaning towards an optimal basis, is developed to provide “good” input.
31 citations