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1.2.2 NGB Interpolation

Using only table values at $x_{k}, x_{k-1}, \dots$ we find

Newton-Gregory Backward interpolation:
$\displaystyle F_{m}(x)$ $\textstyle =$ $\displaystyle \mbox{$f_{k}$}+ \frac{u}{1!} \nabla \mbox{$f_{k}$}+ \frac{u(u+1)}{2!}
{\nabla}^{2} \mbox{$f_{k}$}+ \dots$  
  $\textstyle =$ $\displaystyle \mbox{$f_{k}$}+ \sum_{l=1}^{m} {u+l-1 \choose l}
\nabla^{l} \mbox{$f_{k}$}+ O[(\Delta x)^{m+1}]$  



EXAMPLE: With $m=2$ we arrive at the parabolic NGB approximation

\begin{displaymath}
F_{2}(x) = \mbox{$f_{k}$}+ \frac{\nabla \mbox{$f_{k}$}}{\Del...
...$}}{(\Delta x)^{2}}
(x-x_{k}) (x-x_{k-1})+ O[(\Delta x)^{3}]
\end{displaymath}



Franz J. Vesely Oct 2005
See also:
"Computational Physics - An Introduction," Kluwer-Plenum 2001