User Guide 027

16 Appendix F: Piecewise Lagrangian Polynomial Interpolation

Piecewise polynomials in Lagrange form are given by

$l(x)=\sum _{j=1}^{n+1}y_{j}l_{j}(x)$

(16-1)

where

y_j= interpolation data values corresponding to x_j

n = order or the interpolation polynomial

l_j(x) = Lagrange basis polynomials given by

$l_{j}(x)=\prod _{1<k\leq n+1\ k\neq j}{\frac {x-x_{k}}{x_{j}-x_{k}}}$

(16-2)

The Lagrange basis polynomials are such that

$l_{\neq 1}(x_{i})=0,l_{j=1}=1,l(x_{j})=y_{i}$

( 16-3)

One advantage of using Lagrange polynomials is that the interpolation weights (Lagrange basis functions l_j ) are not a function of y_j . This prop-erty is useful when many interpolations are needed for the same x_j but different y_j such as in the case of interpolating spatial datasets in time.