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Bivariate Thiele-Like Rational Interpolation Continued Fractions with Parameters Based on Virtual Points

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  • Le Zou

    (School of Artificial Intelligence and Big Data, Hefei University, Hefei 230601, China
    Institute of Intelligent Machines, Hefei Institutes of Physical Science, Chinese Academy of Sciences, Hefei 230031, China
    University of Science and Technology of China, Hefei 230026, China)

  • Liangtu Song

    (Institute of Intelligent Machines, Hefei Institutes of Physical Science, Chinese Academy of Sciences, Hefei 230031, China
    University of Science and Technology of China, Hefei 230026, China)

  • Xiaofeng Wang

    (School of Artificial Intelligence and Big Data, Hefei University, Hefei 230601, China)

  • Yanping Chen

    (School of Artificial Intelligence and Big Data, Hefei University, Hefei 230601, China)

  • Chen Zhang

    (School of Artificial Intelligence and Big Data, Hefei University, Hefei 230601, China)

  • Chao Tang

    (School of Artificial Intelligence and Big Data, Hefei University, Hefei 230601, China)

Abstract

The interpolation of Thiele-type continued fractions is thought of as the traditional rational interpolation and plays a significant role in numerical analysis and image interpolation. Different to the classical method, a novel type of bivariate Thiele-like rational interpolation continued fractions with parameters is proposed to efficiently address the interpolation problem. Firstly, the multiplicity of the points is adjusted strategically. Secondly, bivariate Thiele-like rational interpolation continued fractions with parameters is developed. We also discuss the interpolant algorithm, theorem, and dual interpolation of the proposed interpolation method. Many interpolation functions can be gained through adjusting the parameter, which is flexible and convenient. We also demonstrate that the novel interpolation function can deal with the interpolation problems that inverse differences do not exist or that there are unattainable points appearing in classical Thiele-type continued fractions interpolation. Through the selection of proper parameters, the value of the interpolation function can be changed at any point in the interpolant region under unaltered interpolant data. Numerical examples are given to show that the developed methods achieve state-of-the-art performance.

Suggested Citation

  • Le Zou & Liangtu Song & Xiaofeng Wang & Yanping Chen & Chen Zhang & Chao Tang, 2020. "Bivariate Thiele-Like Rational Interpolation Continued Fractions with Parameters Based on Virtual Points," Mathematics, MDPI, vol. 8(1), pages 1-21, January.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:1:p:71-:d:304696
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    References listed on IDEAS

    as
    1. Le Zou & Shuo Tang, 2014. "A New Approach to General Interpolation Formulae for Bivariate Interpolation," Abstract and Applied Analysis, Hindawi, vol. 2014, pages 1-11, June.
    2. Akal, C. & Lukashov, A., 2018. "Newton–Padé approximations for multivariate functions," Applied Mathematics and Computation, Elsevier, vol. 334(C), pages 367-374.
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