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A 5-instant finite difference formula to find discrete time-varying generalized matrix inverses, matrix inverses, and scalar reciprocals

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Abstract

Finite difference schemes have been widely studied because of their fundamental role in numerical analysis. However, most finite difference formulas in the literature are not suitable for discrete time-varying problems because of intrinsic limitations and their relatively low precision. In this paper, a high-precision 1-step-ahead finite difference formula is developed. This 5-instant finite difference (5-IFD) formula is used to approximate and discretize first-order derivatives, and it helps us to compute discrete time-varying generalized matrix inverses. Furthermore, as special cases of generalized matrix inverses, time-varying matrix inversion, and scalar reciprocals are generally deemed as independent problems and studied separately, which are solved unitedly in this paper. The precision of the 5-IFD formula and the convergence behavior of the corresponding discrete-time models are derived theoretically and shown in numerical experiments. Conventional useful formulas, such as the Euler forward finite difference (EFFD) formula and the 4-instant finite difference (4-IFD) formula are also used for comparisons and to show the superiority of the 5-IFD formula.

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Funding

The work is supported by the National Natural Science Foundation of China (with numbers 61473323), by the Foundation of Key Laboratory of Autonomous Systems and Networked Control, Ministry of Education, China (with number 2013A07), and also by the Laboratory Open Fund of Sun Yat-sen University (with number 20160209). Kindly note that all authors of the paper are jointly of the first authorship and jointly of the corresponding authorship.

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Authors and Affiliations

  1. School of Information Science and Technology, Sun Yat-sen University, Guangzhou, 510006, People’s Republic of China

    Jian Li, Mingzhi Mao & Yunong Zhang

  2. Department of Mathematics and Statistics, Auburn University, Auburn, AL, 36849-5310, USA

    Frank Uhlig

Authors
  1. Jian Li
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  2. Mingzhi Mao
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  3. Frank Uhlig
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  4. Yunong Zhang
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Correspondence to Mingzhi Mao.

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Li, J., Mao, M., Uhlig, F. et al. A 5-instant finite difference formula to find discrete time-varying generalized matrix inverses, matrix inverses, and scalar reciprocals. Numer Algor 81, 609–629 (2019). https://doi.org/10.1007/s11075-018-0564-5

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