Multivariate Newton Interpolation
Michael Hecht, Karl B. Hoffmann, Bevan L. Cheeseman, Ivo F. Sbalzarini

TL;DR
This paper extends Newton interpolation to multiple dimensions, introduces an efficient algorithm for polynomial interpolation, and demonstrates that unisolvent Newton-Chebyshev nodes effectively approximate Sobolev functions without Runge's phenomenon.
Contribution
It generalizes Newton interpolation to multivariate spaces, develops the PIP-SOLVER algorithm, and introduces unisolvent Newton-Chebyshev nodes for stable, high-dimensional polynomial approximation.
Findings
The PIP-SOLVER computes solutions in quadratic time relative to node count.
Unisolvent Newton-Chebyshev nodes avoid Runge's phenomenon in high dimensions.
Arbitrary Sobolev functions can be uniformly approximated using the proposed method.
Abstract
For , and a given function , the polynomial interpolation problem (PIP) is to determine a unisolvent node set of points and the uniquely defined polynomial in variables of degree that fits on , i.e., , . For the solution to the PIP is well known. In higher dimensions, however, no closed framework was available. We here present a generalization of the classic Newton interpolation from one-dimensional to arbitrary-dimensional spaces. Further we formulate an algorithm, termed PIP-SOLVER, based on a multivariate divided difference scheme that computes the solution in …
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Taxonomy
TopicsMathematical functions and polynomials · Polynomial and algebraic computation · Advanced Numerical Analysis Techniques
