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A Mixed Precision jacobi SVD algorithm

ACM TRANSACTIONS ON MATHEMATICAL SOFTWARE 2025年第1期51卷 1-33页

作者： Gao, Weiguo Ma, Yuxin Shao, Meiyue Fudan Univ Sch Math Sci Shanghai Peoples R China Fudan Univ Sch Data Sci Shanghai Peoples R China Charles Univ Prague Fac Math & Phys Dept Numer Math Prague Czech Republic Fudan Univ Sch Data Sci Shanghai Peoples R China Fudan Univ MOE Key Lab Computat Phys Sci Shanghai Peoples R China

We propose a mixed precision jacobi algorithm for computing the singular value decomposition (SVD) of a dense matrix. After appropriate preconditioning, the proposed algorithm computes the SVD in a lower precision as an initial guess and then performs one-sided jacobi rotations in the working precision as iterative refinement. By carefully transforming a lower precision solution to a higher precision one, our algorithm achieves about 2x speedup on the x86-64 architecture compared to the usual one-sided jacobi SVD algorithm in LAPACK, without sacrificing the accuracy. CCS Concepts: center dot Mathematics of computing -> Computations on matrices

关键词： Singular value decomposition jacobi algorithm mixed precision algorithm high relative accuracy

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jacobi algorithm for symmetric eigenvalue problem and integrable gradient system of Lax form

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Japan Journal of Industrial and Applied Mathematics 1997年第2期14卷 159-168页

作者： Nakamura, Yoshimasa Department of Electronics Doshisha University Tanabe Kyoto 610-03 Japan Department of Mathematical Science Grad. School of Engineering Science Osaka University Toyonaka 560 Japan

An intimate connection between matrix eigenvalue algorithms and integrable dynamical systems is studied. It is proved that an infinitesimal rotation of each step of the jacobi algorithm for symmetric eigenvalue proble... 详细信息

关键词： Integrable gradient systems jacobi algorithm Lax form Matrix eigenvalue problem

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A sort-jacobi algorithm for semisimple Lie algebras

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LINEAR ALGEBRA AND ITS APPLICATIONS 2009年第1期430卷 155-173页

作者： Kleinsteuber, Martin Univ Wurzburg Dept Math D-97072 Wurzburg Germany

A structure preserving sort-jacobi algorithm for computing eigenvalues or singular values is presented. It applies to an arbitrary semisimple Lie algebra on its (-1)-eigenspace of the Cartan involution. Local quadratic convergence for arbitrary cyclic schemes is shown for the regular case. The proposed method is independent of the representation of the underlying Lie algebra and generalizes well-known normal form problems such as e.g. the symmetric, Hermitian, skew-symmetric, symmetric and skew-symmetric R-Hamiltonian eigenvalue problem and the singular value decomposition. (C) 2008 Elsevier Inc. All rights reserved.

关键词： jacobi algorithm Cartan decomposition Restricted-root space Structured eigenvalue decomposition Structured singular value decomposition Local quadratic convergence Exceptional Lie algebra

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FPGA, GPU, and CPU implementations of jacobi algorithm for eigenanalysis

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JOURNAL OF PARALLEL AND DISTRIBUTED COMPUTING 2016年 96卷 172-180页

作者： Torun, Mustafa U. Yilmaz, Onur Akansu, Ali N. New Jersey Inst Technol Dept Elect & Comp Engn Newark NJ 07102 USA

Parallel implementations of jacobi algorithm for eigenanalysis of a matrix on most commonly used high performance computing (HPC) devices such as central processing unit (CPU), graphics processing unit (GPU), and field-programmable gate array (FPGA) are discussed in this paper. Their performances are investigated and compared. It is shown that CPU, even with multi-threaded implementation, is not a feasible option for large dense matrices. For the GPU implementation, performance impact of the global memory access patterns on the GPU board and the memory coalescing are emphasized. Three memory access methods are proposed. It is shown that one of them achieves 81.6% computational performance improvement over the traditional GPU methods, and it runs 68.5 times faster than a single-threaded CPU for a dense symmetric square matrix of size 1,024. Furthermore, FPGA implementation is presented and its performance running on chips from two major manufacturers are reported. A comparison of GPU and FPGA implementations is quantified and ranked. It is reported that FPGA design delivers the best performance for such a task while GPU is a strong competitor requiring less development effort with superior scalability. We predict that emerging big data applications will benefit from real-time and high performance computing implementations of eigenanalysis for information inference and signal analytics in the future. (C) 2016 Elsevier Inc. All rights reserved.

关键词： Eigenanalysis Principal component analysis Karhunen-Loeve transform jacobi algorithm Chess tournament CORDIC Memory coalescing FPGA GPU CPU

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On the Convergence of a Regularized jacobi algorithm for Convex Optimization

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IEEE TRANSACTIONS ON AUTOMATIC CONTROL 2018年第4期63卷 1113-1119页

作者： Banjac, Goran Margellos, Kostas Goulart, Paul J. Univ Oxford Dept Engn Sci Oxford OX1 3PJ England

In this paper, we consider the regularized version of the jacobi algorithm, a block coordinate descent method for convex optimization with an objective function consisting of the sum of a differentiable function and a block-separable function. Under certain regularity assumptions on the objective function, this algorithm has been shown to satisfy the so-called sufficient decrease condition, and consequently, to converge in objective function value. In this paper, we revisit the convergence analysis of the regularized jacobi algorithm and show that it also converges in iterates under very mild conditions on the objective function. Moreover, we establish conditions under which the algorithm achieves a linear convergence rate.

关键词： Block coordinate descent methods decentralized optimization jacobi algorithm linear convergence

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Application of jacobi algorithm for ISI channels

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IET SIGNAL PROCESSING 2009年第2期3卷 119-132页

作者： Xiao, P. Sellathurai, M. Queens Univ Belfast Inst Elect Commun & Informat Technol Belfast BT3 9DT Antrim North Ireland

Here, the jacobi iterative algorithm is applied to combat intersymbol interference (ISI) caused by frequency-selective channels. The performance bound of the equaliser is analysed in order to gain an insight into its asymptotic behaviour. Because of the error propagation problem, the potential of this algorithm is not reached in an uncoded system. However, its extension to a coded system with the application of the turbo-processing principle results in a new turbo equalisation algorithm, which demonstrates comparable performance with reduced complexity compared with some existing filter-based turbo equalisation schemes;and superior performance compared with some frequency domain solutions, such as orthogonal frequency division multiplexing and single-carrier frequency domain equalisation.

关键词： intersymbol interference jacobi algorithm Electromagnetic compatibility and interference frequency domain solution Modulation and coding methods jacobian matrices single-carrier frequency domain equalisation telecommunication channels ISI channels orthogonal frequency division multiplexing Communication channel equalisation and identification frequency-selective channels jacobi iterative algorithm OFDM modulation turbo equalisation algorithm Algebra equalisers error propagation problem

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An extension of the jacobi algorithm for multi-valued mixed complementarity problems

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OPTIMIZATION 2007年第3期56卷 399-416页

作者： Konnov, I. V. Kazan VI Lenin State Univ Dept Appl Math Kazan 420008 Russia

We consider a generalized mixed complementarity problem (MCP) with box constraints and multi-valued cost mapping. We introduce a concept of an upper Z-mapping, which generalizes the well-known concept of the single- valued Z-mapping and involves the diagonal multi-valued mappings, and suggest an extension of the jacobi algorithm for the above problem containing a composition of such mappings. Being based on its convergence theorem, we establish several existence and uniqueness results. Some examples of the applications are also given.

关键词： mixed complementarity problem multi-valued mappings upper Z-mappings jacobi algorithm existence of solutions

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Analytical Solution of Linear Intersection Based on jacobi algorithm and Its Comparison with Least Squares Solution

Analytical Solution of Linear Intersection Based on Jacobi A...

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3rd International Conference on Civil Engineering, Architecture and Building Materials (CEABM 2013)

作者： Zeng, Huaien China Three Gorges Univ Coll Civil Engn & Architecture Yichang Peoples R China

ISBN: (纸本)9783037857755

The paper introduces the jacobi algorithm, and the jacobi algorithm solution of linear intersection. A numerical cased is illustrated to validate the approach. The result shows the jacobi algorithm solution is approximately equal to least squares (LS) solution, and needs no initial parameter value, and iteration. Thus, it is very important for the nonlinear adjustment, epically for the case that initial parameter value is unknown and hard to be obtained.

关键词： Linear intersection analytical solution jacobi algorithm least squares

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Hardware Architecture for Eigenvalues Computation using the Modified jacobi algorithm on FPGA 5

Hardware Architecture for Eigenvalues Computation using the ...

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5th IEEE International Conference on Signal Processing, Computing and Control (ISPCC)

作者： Gupta, Krishna Deep Wajid, Mohd Muzammil, Rehan Arif, Syed Javed Aligarh Muslim Univ ZHCET Dept Elect Engn Aligarh Uttar Pradesh India

ISBN: (纸本)9781728139883

The calculation of eigenvalues (EVs) is important in the domain of complex computation and processing of matrices of a higher order. For real-time applications, to get superior performance, the architecture of EV computation comes in handy. In this paper, the authors have investigated an efficient architecture for EV computation using the jacobi algorithm on the Xilinx Zed-Board FPGA evaluation platform, Artix-7 family. The iterative algorithm proves to be faster and more efficient in terms of area consumed in the FPGA architecture and also in terms of accuracy of the EV computation. The proposed architecture can take up to matrices of order 20 x 20 using the jacobi algorithm. The algorithm also works for odd order matrices with few modifications in the input matrix. The architecture was implemented on Zynq-7000 xc7z020clg484-1 FPGA and it takes around 4216 LUTs out of 53200 LUTs for matrices of dimensions 4 x 4.

关键词： FPGA Eigenvalues jacobi algorithm Parallel Iterative algorithm

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jacobi's algorithm on compact Lie algebras

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SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS 2004年第1期26卷 42-69页

作者： Kleinsteuber, M Helmke, U Hüper, K Univ Wurzburg Inst Math D-97074 Wurzburg Germany Natl ICT Australia Ltd Syst Engn & Complex Syst Program Canberra ACT 2601 Australia

A generalization of the cyclic jacobi algorithm is proposed that works in an arbitrary compact Lie algebra. This allows, in particular, a unified treatment of jacobi algorithms on different classes of matrices, e. g., skew-symmetric or skew-Hermitian Hamiltonian matrices. Wildberger has established global, linear convergence of the algorithm for the classical jacobi method on compact Lie algebras. Here we prove local quadratic convergence for general cyclic jacobi schemes.

关键词： compact Lie algebras cost function jacobi algorithm optimization quadratic convergence real root space decomposition

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