We present generalizations of the classic Newton and Lagrange interpolation schemes to arbitrary dimensions. The core contribution that enables this new method is the notion of unisolvent nodes, i.e., nodes on which the multivariate polynomial interpolant of a function is unique. We prove that by choosing these nodes in a proper way, the resulting interpolation schemes become generic, while approximating all continuous Sobolev functions. If in addition the function is analytical in the Trefethen domain then, by validation, we achieve the optimal exponential approximation rate given by the upper bound in Trefethen's Theorem. The number of interpolation nodes required for computing the optimal interpolant depends sub-exponentially on the dimension, hence resisting the curse of dimensionality. Based on this, we propose an algorithm that can efficiently and numerically stably solve arbitrarydimensional interpolation problems, and approximate non-analytical functions, with at most quadratic runtime and linear memory requirement.
Many applications in the sciences require numerically stable and computationally efficient evaluation of multivariate polynomials. Finding beneficial representations of polynomials, such as Horner factorisations, is therefore crucial. multivar_horner (Michelfeit, 2018), the Python package presented here, is, as far as we are aware, the first open-source software for computing multivariate Horner factorisations. This paper briefly outlines the functionality of the package and places it in context with respect to previous work in the field. Benchmarks additionally demonstrate the advantages of the implementation and Horner factorisations in general.
Many applications in the sciences require numerically stable and computationally efficient evaluation of multivariate polynomials. Finding beneficial representations of polynomials, such as Horner factorisations, is therefore crucial. multivar horner[1], the python package presented here, is the first open source software for computing multivariate Horner factorisations. This work briefly outlines the functionality of the package and puts it into reference to previous work in the field. Benchmarks additionally prove the advantages of the implementation and Horner factorisations in general.
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