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1.1 Definitions

Given equidistant table values $  f_{k} \equiv f(x_{k})$, define

$\displaystyle \Delta f_{k}$ $\textstyle \equiv$ $\displaystyle f_{k+1}-f_{k} \;\;
{\rm Forward\hspace{1em}Difference}$  
$\displaystyle \nabla f_{k}$ $\textstyle \equiv$ $\displaystyle f_{k}-f_{k-1} \;\;
{\rm Backward \hspace{1ex}Difference}$  
$\displaystyle \delta f_{k}$ $\textstyle \equiv$ $\displaystyle f_{k+1/2}-f_{k-1/2} \;\;\;
{\rm Central}\;\;{\rm Difference}\;\; {\rm (*)}$  



(*) Table values at $x_{k \pm 1/2}$ not given; please have patience! In an emergency, use the ``central mean''
$\displaystyle \delta f_{k} \rightarrow \mu \delta f_{k}$ $\textstyle \equiv$ $\displaystyle \frac{1}{2} \left[ \delta f_{k+1/2}+ \delta f_{k-1/2}\right]$  
  $\textstyle =$ $\displaystyle \frac{1}{2} \left[f_{k+1}-f_{k-1}\right]$  

which uses only table values.


Recursive definition:
$\displaystyle \Delta^{2} f_{k}$ $\textstyle \equiv$ $\displaystyle \Delta f_{k+1}-\Delta f_{k}
= f_{k+2}-2f_{k+1}+f_{k}$  
$\displaystyle \nabla^{2} f_{k}$ $\textstyle \equiv$ $\displaystyle \dots \;\;{\rm (Exercise!?)}$  
$\displaystyle \delta^{2} f_{k}$ $\textstyle \equiv$ $\displaystyle \delta f_{k+1/2}-\delta f_{k-1/2}$  
  $\textstyle =$ $\displaystyle f_{k+1}-2f_{k}+f_{k-1} \;\; {\rm\hspace{1ex}(*)}$  

etcetera.

(*) Here is the reward for your patience!
next up previous
Next: 1.2 Interpolation Formulae Up: 1. Finite Difference Calculus Previous: 1. Finite Difference Calculus
Franz J. Vesely Oct 2005
See also:
"Computational Physics - An Introduction," Kluwer-Plenum 2001