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Acceleration of iterative refinement for singular value decomposition

numerical ALGORITHMS 2024年第2期95卷 979-1009页

作者： Uchino, Yuki Terao, Takeshi Ozaki, Katsuhisa Shibaura Inst Technol Grad Sch Engn & Sci 307 FukasakuMinuma Ku Saitama Saitama 3378570 Japan RIKEN Ctr Computat Sci 7-1-26 Minatojima Minami MachiChuo Ku Kobe Hyogo 6500047 Japan Kyushu Univ Res Inst Informat Technol 744 MotookaNishi Ku Fukuoka 8190395 Japan Shibaura Inst Technol Dept Math Sci 307 FukasakuMinuma Ku Saitama Saitama 3378570 Japan

We propose fast numerical algorithms to improve the accuracy of singular vectors for a real matrix. Recently, Ogita and Aishima proposed an iterative refinement algorithm for singular value decomposition that is constructed with highly accurate matrix multiplications carried out six times per iteration. The algorithm runs for the problem that has no multiple and clustered singular values. In this study, we show that the same algorithm can be run with highly accurate matrix multiplications carried out five times. Also, we proposed four algorithms constructed with highly accurate matrix multiplications, two algorithms with the multiplications carried out four times, and the other two with the multiplications carried out five times. These algorithms adopt the idea of a mixed-precision iterative refinement method for linear systems. numerical experiments demonstrate speed-up and quadratic convergence of the proposed algorithms. As a result, the fastest algorithm is 1.7 and 1.4 times faster than the Ogita-Aishima algorithm per iteration on a CPU and GPU, respectively.

关键词： Singular value decomposition Iterative refinement Mixed-precision computation accurate numerical computation

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High-Performance Eigensolver Combining EigenExa and Iterative Refinement

High-Performance Eigensolver Combining EigenExa and Iterativ...

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2024 Workshops of the International Conference for High Performance Computing, Networking, Storage and Analysis, SC Workshops 2024

作者： Uchino, Yuki Imamura, Toshiyuki RIKEN Center for Computational Science Hyogo Kobe Japan

ISBN: (纸本)9798350355543

This study proposes a high-performance and reliable eigensolver via mixed-precision arithmetic between ordinary and highly-accurate precisions. Eigenvalue decomposition is ubiquitous in simulations. Various eigensolvers for computing approximations have been developed thus far. If eigenvalues are narrowly clustered, the computation of eigenvectors may be ill-posed. Thus, the computed eigenpairs may not be sufficiently accurate and lack reliability. In this study, we introduce mixed-precision iterative refinement methods to improve the accuracy of eigenvectors obtained using numerical methods. This approach contributes to obtaining sufficiently accurate results without arbitrary precision eigensolvers. We construct a high-performance and reliable eigensolver by combining the iterative refinement methods and EigenExa, a modern high-performance solver for large-scale and highly parallel computations. numerical experiment results demonstrate the accuracy of the results and performance benchmark of the proposed approach. © 2024 IEEE.

关键词： accurate numerical computation eigenvalue decomposition iterative refinement

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High-Performance Eigensolver Combining EigenExa and Iterative Refinement 24

High-Performance Eigensolver Combining EigenExa and Iterativ...

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Proceedings of the SC '24 Workshops of the International Conference on High Performance Computing, Network, Storage, and Analysis

作者： Yuki Uchino Toshiyuki Imamura RIKEN Center for Computational Science Kobe Hyogo Japan

ISBN: (纸本)9798350355543

关键词： accurate numerical computation

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Improving Floating-Point Numbers: A Lazy Approach to Adaptive Accuracy Refinement for numerical computations 25th

Improving Floating-Point Numbers: A Lazy Approach to Adaptiv...

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25th European Symposium on Programming (ESOP) Held as Part of the European Joint Conferences on Theory and Practice of Software (ETAPS)

作者： Kawabata, Hideyuki Iwasaki, Hideya Hiroshima City Univ Asa Minami Ku 3-4-1 Ozuka Higashi Hiroshima 7313194 Japan Univ Electrocommun 1-5-1 Chofugaoka Chofu Tokyo 1828585 Japan

ISBN: (纸本)9783662494981;9783662494974

numerical computation using floating-point numbers is prone to accuracy degradation due to round-off errors and cancellation of significant digits. Although multiple-precision arithmetic might alleviate this problem, it is difficult to statically decide the optimal degree of precision for each operation in a program. This paper presents a solution to this problem: the partial results in floating-point representations are incrementally improved using adaptive control. This process of adaptive accuracy refinement is implemented using lazy lists, each one containing a sequence of floating-point numbers with gradually improving accuracies. The computation process is driven by the propagation of demand for more accurate results. The concept of this improving floating-point number (IFN) mechanism was experimentally implemented in two ways: as a Haskell library and as a pure C library. Despite the simple approach, the results for numerical problems demonstrated the effectiveness of this mechanism.

关键词： Improving floating-point numbers accurate numerical computation Lazy evaluation Haskell library

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