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SIAM REVIEW 2008年第1期50卷 133-145页

作者： Watkins, David S. Washington State Univ Dept Math Pullman WA 99164 USA

The qr algorithm is still one of the most important methods for computing eigenvalues and eigenvectors of matrices. Most discussions of the qr algorithm begin with a very basic version and move by steps toward the versions of the algorithm that are actually used. This paper outlines a pedagogical path that leads directly to the implicit multishift qr algorithms that are used in practice, bypassing the basic qr algorithm completely.

关键词： eigenvalue eigenvector qr algorithm Hessenberg form

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Reduced Complexity Maximum-Likelihood Detection for MIMO-OFDM Systems

Reduced Complexity Maximum-Likelihood Detection for MIMO-OFD...

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8th International Conference on Wireless Communications, Networking and Mobile Computing (WiCOM)

作者： You, Weizhi Yi, Lilin Hu, Weisheng Shanghai Jiao Tong Univ Dept Elect Engn State Key Lab Adv Opt Commun Syst & Networks Shanghai 200240 Peoples R China

ISBN: (纸本)9781612846835;9781612846828

A reduced complexity Maximum-Likelihood (ML) detection algorithm is proposed for the multiple-input multiple-output orthogonal frequency division multiple (MIMO-OFDM) systems. The proposed detection algorithm combines the ML algorithm and the qr algorithm. In the detection process, the first T signals are detected by the ML algorithm and the last Nt-T signals are detected by qr algorithm where T is a parameter and Nt is the number of transmitter antennas. From the simulation results, compared with the traditional ML algorithm, the computational complexity of the proposed algorithm with 4 transmitter antennas, 4 receiver antennas and T=3 is reduced by 95% at the expense of about 1.3dB signal-to-noise-ratio (SNR) degradation for bit error rate (BER) at 10(-3). Therefore, the proposed detection algorithm can be used in the practical MIMO-OFDM systems requiring very low complexity.

关键词： MIMO-OFDM qr-decomposition qr algorithm ML algorithm

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qr methods and error analysis for computing Lyapunov and Sacker-Sell spectral intervals for linear differential-algebraic equations

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ADVANCES IN COMPUTATIONAL MATHEMATICS 2011年第2-4期35卷 281-322页

作者： Vu Hoang Linh Mehrmann, Volker Van Vleck, Erik S. Tech Univ Berlin Inst Math D-10623 Berlin Germany Vietnam Natl Univ Fac Math Mech & Informat Hanoi Vietnam Univ Kansas Dept Math Lawrence KS 66045 USA

In this paper, we propose and investigate numerical methods based on qr factorization for computing all or some Lyapunov or Sacker-Sell spectral intervals for linear differential-algebraic equations. Furthermore, a perturbation and error analysis for these methods is presented. We investigate how errors in the data and in the numerical integration affect the accuracy of the approximate spectral intervals. Although we need to integrate numerically some differential-algebraic systems on usually very long time-intervals, under certain assumptions, it is shown that the error of the computed spectral intervals can be controlled by the local error of numerical integration and the error in solving the algebraic constraint. Some numerical examples are presented to illustrate the theoretical results.

关键词： Differential-algebraic equation Strangeness index Lyapunov exponent Sacker-Sell spectrum Exponential dichotomy Spectral interval Smooth qr factorization qr algorithm Kinematic equivalence Steklov function

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From qd to LR, or, how were the qd and LR algorithms discovered?

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IMA JOURNAL OF NUMERICAL ANALYSIS 2011年第3期31卷 741-754页

作者： Gutknecht, Martin H. Parlett, Beresford N. ETH Seminar Appl Math CH-8092 Zurich Switzerland Univ Calif Berkeley Dept Math Berkeley CA 94720 USA

Perhaps, the most astonishing idea in eigenvalue computation is Rutishauser's idea of applying the LR transform to a matrix for generating a sequence of similar matrices that become more and more triangular. The same idea is the foundation of the ubiquitous qr algorithm. It is well known that this idea originated in Rutishauser's qd algorithm, which precedes the LR algorithm and can be understood as applying LR to a tridiagonal matrix. But how did Rutishauser discover qd and when did he find the qd-LR connection? We checked some of the early sources and have come up with an explanation.

关键词： matrix eigenvalues qd algorithm LR algorithm qr algorithm Rutishauser

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Continuous spectra and numerical eigenvalues

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MATHEMATICAL AND COMPUTER MODELLING 2011年第11-12期54卷 2616-2622页

作者： Guba, Oksana Lorenz, Jens Sandia Natl Labs Albuquerque NM 87185 USA Univ New Mexico Dept Math & Stat Albuquerque NM 87131 USA

Some spectral problems for differential operators are naturally posed on the whole real line, often leading to eigenvalues plus continuous spectrum. Then the numerical approximation typically involves three processes: (a) reduction to a finite interval;(b) discretization;(c) application of a numerical eigenvalue solver such as the qr-algorithm. Reduction to a finite interval and discretization typically eliminate the continuous spectrum. However, through round-off error, the continuous spectrum may show up again when the eigenvalue solver is applied. (In some sense, three wrongs make a right.) Interestingly, not all parts of the continuous spectrum show up in the same way, however. We illustrate this observation by numerical examples. A perturbation argument, though non-rigorous, explains the observation. (C) 2011 Elsevier Ltd. All rights reserved.

关键词： Eigenvalue problems Continuous spectrum qr algorithm

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On iterative qr pre-processing in the parallel block-Jacobi SVD algorithm

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PARALLEL COMPUTING 2010年第5-6期36卷 297-307页

作者： Becka, Martin Oksa, Gabriel Vajtersic, Marian Grigori, Laura Slovak Acad Sci Inst Math Dept Informat Bratislava Slovakia Salzburg Univ Dept Comp Sci A-5020 Salzburg Austria Univ Paris 11 INRIA Orsay France

An efficient version of the parallel two-sided block-Jacobi algorithm for the singular value decomposition of an m x n matrix A includes the pre-processing step, which consists of the qr factorization of A with column pivoting followed by the optional LQ factorization of the R-factor. Then the iterative two-sided block-Jacobi algorithm is applied in parallel to the R-factor (or L-factor). For the efficient computation of the parallel qr (or LQ) factorization with (or without) column pivoting implemented in the ScaLAPACK, some matrix block cyclic distribution on a process grid r x c with p = r x c, r,c >= 1, and block size n(b) x n(b) is required so that all processors remain busy during the whole parallel qr (or LQ) factorization. Optimal values for parameters r, c and nb are estimated experimentally using matrices of order n = 4000 and 8000, and the number of processors p = 8 and 16, respectively. It turns out that the optimal values are about n(b) = 100 and r <= c with both r, c near to root p. These parameters are then used in numerical experiments for six various distributions of singular values combined with well- (k = 10(1)) and ill-conditioned matrices (K = 108). It is shown that using optimal parameters in the pre-processing step, the parallel two-sided block-Jacobi SVD algorithm performs better (or equally well) than the ScaLAPACK routine PDGESVD for matrices with a multiple minimal/maximal singular value regardless to the condition number. For other distributions of singular values, our algorithm is slower than the ScaLAPACK. The un-pivoted qrLQ pre-processing step is then re-formulated and extended to the qr iteration, and its connection to the qr algorithm applied to specific symmetric, positive definite matrices is shown. This connection helps to explain observations in another set of experiments with a variable number of qr iteration steps. In general, the best results for all six distributions of singular values are achieved by using about six qr iter

关键词： qr iteration qr algorithm Singular value decomposition Blocking factor Process grid Cyclic matrix distribution Two-sided block-Jacobi method Message passing interface

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The periodic qr algorithm is a disguised qr algorithm

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LINEAR ALGEBRA AND ITS APPLICATIONS 2006年第2-3期417卷 423-433页

作者： Kressner, Daniel Tech Univ Berlin Inst Math MA 4 5 D-10623 Berlin Germany

The periodic qr algorithm is a strongly backward stable method for computing the eigenvalues of products of matrices, or equivalently for computing the eigenvalues of block cyclic matrices. The main purpose of this paper is to show that this algorithm is numerically equivalent to the standard qr algorithm. It will be demonstrated how this connection may be used to develop a better understanding of the periodic qr algorithm. (c) 2003 Elsevier Inc. All rights reserved.

关键词： qr algorithm block cyclic matrices matrix products

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Shifting algorithms with Maple and implicit shift in the qr algorithm

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APPLIED MATHEMATICS AND COMPUTATION 2005年第3期161卷 947-962页

作者： Najafi, HS Shams, M Guilan Univ Fac Sci Dept Math Rasht Iran Guilan Univ Ctr Comp Rasht Iran

We have written out all shifting algorithm in the qr corresponding to article [SIAM J. Matrix Anal. 12 (1991) 385], specially the block shift algorithm, and tested them with Maple8. A comparison table has been designed in each case to compare the implementation of the algorithms with each other. The differences of the shifts have been noted in the last section. (C) 2004 Elsevier Inc. All rights reserved.

关键词： eigenvalue problem qr algorithm implicit and explicit shift block Hessenberg matrix block shift

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Performance and accuracy of LAPACK's symmetric tridiagonal eigensolvers

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SIAM JOURNAL ON SCIENTIFIC COMPUTING 2008年第3期30卷 1508-1526页

作者： Demmel, James W. Marques, Osni A. Parlett, Beresford N. Voemel, Christof Univ Calif Berkeley Dept Math Berkeley CA 94720 USA Univ Calif Berkeley Div Comp Sci Berkeley CA 94720 USA Univ Calif Berkeley Computat Res Div Berkeley CA 94720 USA

We compare four algorithms from the latest LAPACK 3.1 release for computing eigenpairs of a symmetric tridiagonal matrix. These include qr iteration, bisection and inverse iteration (BI), the divide-and-conquer method (DC), and the method of multiple relatively robust representations (MR). Our evaluation considers speed and accuracy when computing all eigenpairs and additionally subset computations. Using a variety of carefully selected test problems, our study includes a variety of today's computer architectures. Our conclusions can be summarized as follows. (1) DC and MR are generally much faster than qr and BI on large matrices. (2) MR almost always does the fewest floating point operations, but at a lower MFlop rate than all the other algorithms. (3) The exact performance of MR and DC strongly depends on the matrix at hand. (4) DC and qr are the most accurate algorithms with observed accuracy O(root n epsilon). The accuracy of BI and MR is generally O(ne). (5) MR is preferable to BI for subset computations.

关键词： LAPACK symmetric eigenvalue problem inverse iteration divide and conquer qr algorithm multiple relatively robust representations algorithm accuracy performance benchmark

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Spectral Rayleigh-Schrodinger series revisited

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NUMERICAL FUNCTIONAL ANALYSIS AND OPTIMIZATION 2008年第7-8期29卷 735-749页

作者： Ahues, Mario Largillier, Alain LaMUSE F-42100 St Etienne France

Rayleigh-Schrodinger series for perturbation bounds of spectral elements is revisited. The convergence radius is estimated for bases of spectral subspaces. Applications to both Hessenberg and Hermitian matrices are de... 详细信息

关键词： eigenvalues eigenvectors qr algorithm Rayleigh-Schrodinger series spectral subspaces

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