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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
DOI: https://doi.org/10.1090/S0025-5718-1995-1297477-5
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.


References [Enhancements On Off] (What's this?)

  • [1] C. de Boor and A. Ron, Computational aspects of polynomial interpolation in several variables, Math. Comp. 58 (1992), 705-727. MR 1122061 (92i:65022)
  • [2] M. Gasea, Multivariate polynomial interpolation Computation of Curves and Surfaces (W. Dahmen, M. Gasea, and C. A. Micchelli, eds.), Kluwer, Dordrecht, 1990, pp. 215-236. MR 1064962 (91k:65028)
  • [3] E. Isaacson and H. B. Keller, Analysis of numerical methods, Wiley, New York, 1966. MR 0201039 (34:924)
  • [4] R. A. Lorentz, Multivariate Birkhoff interpolation, Lecture Notes in Math., vol. 1516, Springer-Verlag, Berlin and New York, 1992. MR 1222648 (94h:41001)
  • [5] C. A. Micchelli, On a numerically efficient method of computing multivariate B-splines, Multivariate Approximation Theory (W. Schempp and K. Zeller, eds.), Birkhäuser, Basel, 1979, pp. 211-248. MR 560673 (81g:65017)

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Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1995-1297477-5
Keywords: Lagrange interpolation, finite difference, simplex spline, remainder formula, algorithm
Article copyright: © Copyright 1995 American Mathematical Society

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