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SHIFTING STRATEGIES FOR THE PARALLEL qr algorithm

SIAM JOURNAL ON SCIENTIFIC COMPUTING 1994年第4期15卷 953-958页

作者： WATKINS, DS WASHINGTON STATE UNIV DEPT PURE & APPL MATH PULLMAN WA 99164 USA

The use of high-order generalized Rayleigh quotient shifting strategies is advocated as a means of improving the parallel performance of the double-shift qr algorithm for nonsymmetric eigenvalue problems.

关键词： qr algorithm PARALLEL IMPLEMENTATION SHIFTS

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A HAMILTONIAN qr algorithm

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SIAM JOURNAL ON SCIENTIFIC AND STATISTICAL COMPUTING 1986年第1期7卷 212-229页

作者： BYERS, R

This paper presents a variant $qr$ algorithm for calculating a Hamiltonian–Schur decomposition [10]. It is defined for Hamiltonian matrices that arise from single input control systems. Numerical stability and Hamiltonian structure are preserved by using unitary symplectic similarity transformations. Following a finite step reduction to a Hessenberg-like condensed form, a sequence of similarity transformations yields a permuted triangular matrix. As the iteration converges, it deflates into problems of lower dimension. Convergence is accelerated by varying a scalar shift. When the Hamiltonian matrix is real, complex arithmetic can be avoided by using an implicit double shift technique. The Hamiltonian-Schur decomposition yields the same invariant subspace information as a Schur decomposition but requires significantly less work and storage for problems of size greater than about 20.

关键词： qr algorithm Hamiltonian matrices symplectic matrices algebraic Riccati equation optimal control

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Convergence of the unitary qr algorithm with a unimodular Wilkinson shift

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Mathematics of Computation 2002年第241期72卷 375-385页

作者： Wang, Tai-Lin Gragg, William B. Department of Mathematical Sciences National Chengchi University Taipei Taiwan Department of Mathematics Naval Postgraduate School Monterey CA 93943 United States

In applying the qr algorithm to compute the eigenvalues of a unitary Hessenberg matrix, a projected Wilkinson shift of unit modulus is proposed and proved to give global convergence with (at least) a quadratic asymptotic rate for the qr iteration. Experimental testing demonstrates that the unimodular shift produces more efficient numerical convergence. © 2002 American Mathematical Society.

关键词： qr algorithm Shift strategy Unitary Hessenberg matrices

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The qrPS algorithm: A generalization of the qr algorithm for the singular values decomposition of rectangular matrices

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Periodica Mathematica Hungarica 1985年第3期16卷 201-207页

作者： Bolla, M. Tusnády, G. MTA Szamitastechnikai es Automatizalasi Kutato Intezet Budapest H-1502 Kende U. 13-17 Hungary MTA Matematikai Kutató Intézet Budapest H-1364 Reáltanoda U. 13-15 Hungary

A generalization of the qr algorithm proposed by Francis [2] for square matrices is introduced for the singular values decomposition of arbitrary rectangular matrices. Geometrically the algorithm means the subsequent orthogonalization of the image of orthonormal bases produced in the course of the iteration. Our purpose is to show how to get a series of lower triangular matrices by alternate orthogonal-upper triangular decompositions in different dimensions and to prove the convergence of this series. © 1985 Akadémiai Kiadó.

关键词： AMS (MOS) subject classifications (1980): Primary 65F15, Secondary 65F25 qr algorithm Singular values decomposition

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The qr algorithm and hymans method on vector computers

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Mathematics of Computation 1976年第133期30卷 132-142页

作者： Ward, Robert C. Union Carbide Corporation Nuclear Division Computer Sciences Division United States

The implementation on vector computers of the qr algorithm and of iterative schemes based on obtaining the determinant and its derivatives by Hymans method are presented. It is shown that iterative schemes based on Hymans method will probably be more efficient than the qr algorithm on vector computers for large matrices. A theoretical comparison of the Laguerre iterative scheme with the qr algorithm is presented using the latest available CDC STAR-100 instruction execution times. In addition, the results of several test cases run on the Laguerre-Hyman algorithm on a serial computer are reported. © 1976, American Mathematical Society.

关键词： Eigenvalues Hymans method qr algorithm Vector computers

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qr algorithm with two-sided Rayleigh quotient shifts

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NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS 2023年第5期30卷 e2487-e2487页

作者： Chen, Xiao Shan Xu, Hongguo South China Normal Univ Sch Math Guangzhou Guangdong Peoples R China Univ Kansas Dept Math Lawrence KS USA Univ Kansas Dept Math Lawrence KS 66045 USA

We introduce the two-sided Rayleigh quotient shift to the qr algorithm for non-Hermitian matrices to achieve a cubic local convergence rate. For the singly shifted case, the two-sided Rayleigh quotient iteration is incorporated into the qr iteration. A modified version of the method and its truncated version are developed to improve the efficiency. Based on the observation that the Francis double-shift qr iteration is related to a 2D Grassmann-Rayleigh quotient iteration, A doubly shifted qr algorithm with the two-sided 2D Grassmann-Rayleigh quotient double-shift is proposed. A modified version of the method and its truncated version are also developed. Numerical examples are presented to show the convergence behavior of the proposed algorithms. Numerical examples also show that the truncated versions of the modified methods outperform their counterparts including the standard Rayleigh quotient single-shift and the Francis double-shift.

关键词： local convergence rate qr algorithm qr iteration two-sided 2D Grassmann-Rayleigh quotient double-shift two-sided Rayleigh quotient shift

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Comparison of V-BLAST/OSIC algorithm and the qr Decomposition algorithm

Comparison of V-BLAST/OSIC Algorithm and the QR Decompositio...

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2nd International Conference on Measurement, Information and Control (ICMIC)

作者： Zhang Xiuyan Zhang Min Zhao Qingfeng Pi Boyuan Liu Qingqiang Northeast Petr Univ Elect & Informat Engn Coll Daqing 163318 Peoples R China Daqing Oilfield Ltd Co Oil Prod Co 1 Daqing 163001 Peoples R China Daqing Oilfiled Daqing OilField Thermal Power Plant Daqing 163314 Peoples R China

ISBN: (纸本)9781479913909

MIMO-OFDM has become the core of future communication technologies for its great excellence, the detection algorithms are various, but only nonlinear algorithms are extensive used in practice. First, the ZF and MMSE criteria are briefly introduced, the nonlinear detection algorithms of V-BLAST algorithm and qr algorithm are analyzed in detail. Then the algorithms based on ZF criterion and MMSE criterion named ZF-OSIC, MMSE-OSIC, SqrD-ZF and SqrD-MMSE are studied. Finally the BER performance of them are compared, through MATLAB simulation.

关键词： V-BLAST algorithm qr algorithm MIMO-OFDM

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algorithm 1019: A Task-based Multi-shift qr/QZ algorithm with Aggressive Early Deflation

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ACM TRANSACTIONS ON MATHEMATICAL SOFTWARE 2022年第1期48卷 11-11页

作者： Myllykoski, Mirko Umea Univ Dept Comp Sci SE-90187 Umea Sweden Umea Univ HPC2N SE-90187 Umea Sweden

The qr algorithm is one of the three phases in the process of computing the eigenvalues and the eigenvectors of a dense nonsymmetric matrix. This paper describes a task-based qr algorithm for reducing an upper Hessenberg matrix to real Schur form. The task-based algorithm also supports generalized eigenvalue problems (QZ algorithm) but this paper concentrates on the standard case. The task-based algorithm adopts previous algorithmic improvements, such as tightly-coupled multi-shifts and Aggressive Early Deflation (AED), and also incorporates several new ideas that significantly improve the performance. This includes, but is not limited to, the elimination of several synchronization points, the dynamic merging of previously separate computational steps, the shortening and the prioritization of the critical path, and experimental GPU support. The task-based implementation is demonstrated to be multiple times faster than multi-threaded LAPACK and ScaLAPACK in both single-node and multi-node configurations on two different machines based on Intel and AMD CPUs. The implementation is built on top of the StarPU runtime system and is part of the open-source StarNEig library.

关键词： Eigenvalue problem real Schur form qr algorithm QZ algorithm multi-shift aggressive early deflation task-based StarPU shared memory distributed memory MPI GPU

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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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Pole-Swapping algorithms for the Eigenvalue Problem

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丛书名： SIAM Spotlights

2025年

作者： Daan Camps Thomas Mach Raf Vandebril David S. Watkins

Get 10% off by pre-ordering this book.This item has not yet published. Pre-order now and we will ship and process payment when the book becomes available.To pre-order contact Customer Service:800-447-SIAM (US &amp... 详细信息

ISBN: (数字)9781611978377

ISBN: (纸本)9781611978360

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Publishing May 2025

Matrix eigenvalue problems arise in a wide variety of fields in science and engineering, so it is important to have reliable and efficient methods for solving them. Of the methods devised, bulge-chasing algorithms, such as the famous qr and QZ algorithms, are the most important. This book focuses on pole-swapping algorithms, a new class of methods that are generalizations of bulge-chasing algorithms and a bit faster and more accurate owing to their inherent flexibility. The pole-swapping theory developed by the authors sheds light on the functioning of the whole class of algorithms, including qr and QZ.

Pole-Swapping algorithms for the Eigenvalue Problem

is the only book on the topic,
describes the state of the art on eigenvalue methods, and
provides an improved understanding and explanation of why these important algorithms work.

关键词： Eigenvalue problems QZ algorithm qr algorithm bulge chasing pole swapping core transformations

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