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An algorithm for approximating implicit functions by polynomials without using higher order differentiability information

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Abstract

We consider an equation of multiple variables in which a partial derivative does not vanish at a point. The implicit function theorem provides a local existence and uniqueness of the function for the equation. In this paper, we propose an algorithm to approximate the function by a polynomial without using higher order differentiability information, which depends essentially on integrability. Moreover, we extend the method to a system of equations if the Jacobian determinant does not vanish. This is a robust method for implicit functions that are not differentiable to higher order. Additionally, we present two numerical experiments to verify the theoretical results.

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The datasets are generated and analyzed during the current study.

Code Availability

It is available from the corresponding author upon reasonable request.

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Acknowledgements

The author emphasizes that the quality of this paper has been improved by the careful and valuable comments from anonymous reviewers. The author would like to express deep gratitude to the anonymous reviewers.

Funding

This work was supported by the National Research Foundation of Korea (No. 17R1E1A1A03070307).

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Authors and Affiliations

  1. Department of Mathematics, Sogang University, 35 Baekbeom-ro, Mapo-gu, Seoul, 04107, Korea

    Kyung Soo Rim

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  1. Kyung Soo Rim
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Kyung Soo Rim contributed to mathematical modeling, code development, conceptualization, and literature survey.

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Correspondence to Kyung Soo Rim.

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Rim, K.S. An algorithm for approximating implicit functions by polynomials without using higher order differentiability information. Numer Algor (2024). https://doi.org/10.1007/s11075-024-01833-9

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