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Fast multipole methods for approximating a function from sampling values

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Abstract

Both barycentric Lagrange interpolation and barycentric rational interpolation are thought to be stable and effective methods for approximating a given function on some special point sets. A direct evaluation of these interpolants due to N interpolation points at M sampling points requires \(\mathcal {O}(NM)\) arithmetic operations. In this paper, we introduce two fast multipole methods to reduce the complexity to \(\mathcal {O}(\max \left \{N,M\right \})\). The convergence analysis is also presented in this paper.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Central South University, Changsha, Hunan, 410083, People’s Republic of China

    Guidong Liu & Shuhuang Xiang

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  1. Guidong Liu
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  2. Shuhuang Xiang
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Correspondence to Guidong Liu.

Additional information

This work is supported partly by NSF of China (No.11371376), the Innovation-Driven Project and the Mathematics and Interdisciplinary Sciences Project of Central South University.

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Liu, G., Xiang, S. Fast multipole methods for approximating a function from sampling values. Numer Algor 76, 727–743 (2017). https://doi.org/10.1007/s11075-017-0279-z

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  • DOI: https://doi.org/10.1007/s11075-017-0279-z

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