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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 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.

**[1]**Carl de Boor and Amos Ron,*Computational aspects of polynomial interpolation in several variables*, Math. Comp.**58**(1992), no. 198, 705–727. MR**1122061**, 10.1090/S0025-5718-1992-1122061-0**[2]**M. Gasca,*Multivariate polynomial interpolation*, Computation of curves and surfaces (Puerto de la Cruz, 1989) NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 307, Kluwer Acad. Publ., Dordrecht, 1990, pp. 215–236. MR**1064962**, 10.1098/rspa.1968.0185**[3]**Eugene Isaacson and Herbert Bishop Keller,*Analysis of numerical methods*, John Wiley & Sons, Inc., New York-London-Sydney, 1966. MR**0201039****[4]**Rudolph A. Lorentz,*Multivariate Birkhoff interpolation*, Lecture Notes in Mathematics, vol. 1516, Springer-Verlag, Berlin, 1992. MR**1222648****[5]**Charles A. Micchelli,*On a numerically efficient method for computing multivariate 𝐵-splines*, Multivariate approximation theory (Proc. Conf., Math. Res. Inst., Oberwolfach, 1979) Internat. Ser. Numer. Math., vol. 51, Birkhäuser, Basel-Boston, Mass., 1979, pp. 211–248. MR**560673**

Retrieve articles in *Mathematics of Computation*
with MSC:
41A63,
41A05,
65D05

Retrieve articles in all journals with MSC: 41A63, 41A05, 65D05

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