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PACF: A precision-adjustable computational framework for solving singular values

Author

Listed:
  • Li, Chuanying
  • Barrio, Roberto
  • Xiao, Xiong
  • Du, Peibing
  • Jiang, Hao
  • Quan, Zhe
  • Li, Kenli

Abstract

Singular value decomposition (SVD) plays a significant role in matrix analysis, and the differential quotient difference with shifts (DQDS) algorithm is an important technique for solving singular values of upper bidiagonal matrices. However, ill-conditioned matrices and large-scale matrices may cause inaccurate results or long computation times when solving singular values. At the same time, it is difficult for users to effectively find the desired solution according to their needs. In this paper, we design a precision-adjustable computational framework for solving singular values, named PACF. In our framework, the same solution algorithm contains three options: original mode, high-precision mode, and mixed-precision mode. The first algorithm is the original version of the algorithm. The second algorithm is a reliable numerical algorithm we designed using Error-free transformation (EFT) technology. The last algorithm is an efficient numerical algorithm we developed using the mixed-precision idea. Our PACF can add different solving algorithms for different types of matrices, which are universal and extensible. Users can choose different algorithms to solve singular values according to different needs. This paper implements the high-precision DQDS and mixed-precision DQDS algorithms and conducts extensive experiments on a supercomputing platform to demonstrate that our algorithm is reliable and efficient. Besides, we introduce the error analysis of the inner loop of the DQDS and HDQDS algorithms.

Suggested Citation

  • Li, Chuanying & Barrio, Roberto & Xiao, Xiong & Du, Peibing & Jiang, Hao & Quan, Zhe & Li, Kenli, 2023. "PACF: A precision-adjustable computational framework for solving singular values," Applied Mathematics and Computation, Elsevier, vol. 440(C).
    • Handle: RePEc:eee:apmaco:v:440:y:2023:i:c:s0096300322006841
    DOI: 10.1016/j.amc.2022.127611
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    References listed on IDEAS

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    1. Du, Peibing & Barrio, Roberto & Jiang, Hao & Cheng, Lizhi, 2017. "Accurate quotient-difference algorithm: Error analysis, improvements and applications," Applied Mathematics and Computation, Elsevier, vol. 309(C), pages 245-271.
    2. Graillat, Stef & Jézéquel, Fabienne & Picot, Romain, 2018. "Numerical validation of compensated algorithms with stochastic arithmetic," Applied Mathematics and Computation, Elsevier, vol. 329(C), pages 339-363.
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    1. Graillat, Stef & Jézéquel, Fabienne & Picot, Romain, 2018. "Numerical validation of compensated algorithms with stochastic arithmetic," Applied Mathematics and Computation, Elsevier, vol. 329(C), pages 339-363.

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