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Efficient computational method for matrix function in dynamic problems

动力学问题中矩阵函数的高效算法

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Abstract

An algorithm based on the Paterson-Stockmeyer (PS) scheme and filtering technology is developed to compute the large matrix functions in dynamic problems accurately and efficiently. With the assistance of analysis on truncation error and error caused by filtering, the error growth law during the computation is studied, based on which an adaptive filtering threshold is proposed to help the proposed algorithm more efficiently achieve similar accuracy as the original PS scheme. Numerical examples, including 30 random matrices with different bandwidths, 10 adjacency matrices in the complex network dynamic problems, and a trampoline vibration problem, are given to verify the efficiency and accuracy of the proposed algorithm. Numerical results suggest that the proposed algorithm can achieve good accuracy and efficiency in computing the matrix function in the considered dynamic problems.

摘要

本文基于Paterson-Stockmeyer (PS)算法和过滤技术提出了一种能精确和高效计算动力学问题中矩阵函数的算法. 借助对截断误差和过滤引误差的分析, 探究了计算中的误差增长规律, 并基于此给出了自适应过滤阈值, 使得提出的算法能更加高效地达到与原PS算法相似的精度. 数值实验采用了包括30个不同带宽的随机矩阵, 10个复杂网络动力学问题中的邻接矩阵来验证提出算法的效率与精度. 数值实验的结果表明, 提出的算法在计算所考虑的动力学问题的矩阵函数时能获得良好的精度和效率.

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Acknowledgements

This work was supported by the National Natural Science Foundation of China grants (Grant Nos. U1906233, 11472076 and 51609034), the National Key R&D Program of China (Grant No. 2021YFA1003501), the Science Foundation of Liaoning Province of China (Grant No. 2021-MS-119), the Dalian Youth Science and Technology Star project (Grant No. 2018RQ06), and the Fundamental Research Funds for the Central Universities grant (Grant No. DUT20GJ216).

Author information

Authors and Affiliations

  1. Department of Mechanics, State Key Laboratory of Structural Analysis of Industrial Equipment, Dalian University of Technology, Dalian, 116023, China

    Feng Wu  (吴锋), Li Zhu  (朱力), Yuelin Zhao  (赵悦琳), Jun Yan  (阎军) & Wanxie Zhong  (钟万勰)

  2. State Key Laboratory for Turbulence & Complex Systems, Department of Mechanics and Engineering Science, College of Engineering, Peking University, Beijing, 100871, China

    Kailing Zhang  (章恺琳)

  3. Yunnan Machinery Research and Design Institute, China National Machinery Institute Group, Kunming, 650000, China

    Qinghua Shi  (施庆华)

Authors
  1. Feng Wu  (吴锋)
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  2. Li Zhu  (朱力)
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  3. Yuelin Zhao  (赵悦琳)
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  4. Kailing Zhang  (章恺琳)
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  5. Jun Yan  (阎军)
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  6. Wanxie Zhong  (钟万勰)
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  7. Qinghua Shi  (施庆华)
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Contributions

Wu Feng designed the research. Wu Feng, Zhu Li, Zhao Yuelin and Zhang Kailing wrote the first draft of the manuscript. Wu Feng and Zhu Li set up the experiment set-up and processed the experiment data. Wu Feng and Zhu Li helped organize the manuscript. Wu Feng and Zhu Li revised and edited the final version. Yan Jun, Zhong Wanxie and Shi Qinghua helped the funding acquisition for the research. Zhong Wanxie helped the conceptualization for the research.

Corresponding authors

Correspondence to Wanxie Zhong  (钟万勰) or Qinghua Shi  (施庆华).

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Wu, F., Zhu, L., Zhao, Y. et al. Efficient computational method for matrix function in dynamic problems. Acta Mech. Sin. 39, 522451 (2023). https://doi.org/10.1007/s10409-023-22451-x

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