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Mathematics of Computation

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On multivariate Lagrange interpolation

Authors: Thomas Sauer and Yuan Xu
Journal: Math. Comp. 64 (1995), 1147-1170
MSC: Primary 41A63; Secondary 41A05, 65D05
MathSciNet review: 1297477
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Abstract: Lagrange interpolation by polynomials in several variables is studied through a finite difference approach. We establish an interpolation formula analogous to that of Newton and a remainder formula, both of them in terms of finite differences. We prove that the finite difference admits an integral representation involving simplex spline functions. In particular, this provides a remainder formula for Lagrange interpolation of degree n of a function f, which is a sum of integrals of certain $ (n + 1)$st directional derivatives of f multiplied by simplex spline functions. We also provide two algorithms for the computation of Lagrange interpolants which use only addition, scalar multiplication, and point evaluation of polynomials.

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Keywords: Lagrange interpolation, finite difference, simplex spline, remainder formula, algorithm
Article copyright: © Copyright 1995 American Mathematical Society