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Next: 1.3.2 Second Derivatives Up: 1.3 Difference Quotients Previous: 1.3 Difference Quotients

1.3.1 First Derivatives



Replacing $dx$ by $\Delta x$ and $df$ by $\Delta f_{k}$, $\nabla f_{k}$, or $\delta f_{k}$ we arrive at various approximations to the first derivative of $f$ at $x_{k}$:

DNGF
(Differentiated Newton-Gregory Forward):

$\displaystyle {F_{k}}'$ $\textstyle \approx$ $\displaystyle \frac{1}{\Delta x} \left[ \Delta \mbox{$f_{k}$}
- \frac{\Delta^{2} \mbox{$f_{k}$}}{2} + \frac{\Delta^{3} \mbox{$f_{k}$}}{3}
- \dots \right]$  



Example:
$\displaystyle {F_{k}}'$ $\textstyle =$ $\displaystyle \frac{1}{\Delta x}\left[ \Delta f_{k}-\frac{\Delta^{2}f_{k}}{2
}
\right] +O[(\Delta x)^{2}]$  
  $\textstyle =$ $\displaystyle \frac{1}{\Delta x} \left[ -\frac{1}{2}f_{k+2}+2f_{k+1}-\frac{3}{2}f_{k}
\right]+ O[(\Delta x)^{2}]$  




DNGB
(Differentiated Newton-Gregory Backward):

$\displaystyle {F_{k}}'$ $\textstyle \approx$ $\displaystyle \frac{1}{\Delta x}
\left[ \nabla f_{k}
+ \frac{\nabla^{2} f_{k}}{2} + \frac{\nabla^{3} f_{k}}{3}
+ \dots \right]$  



Example:
$\displaystyle {F_{k}}'$ $\textstyle =$ $\displaystyle \frac{1}{\Delta x}
\left[ \nabla f_{k}+\frac{\nabla^{2}f_{k}}{2} \right]
+ O[(\Delta x)^{2}]$  
  $\textstyle =$ $\displaystyle \frac{1}{\Delta x} \left[ \frac{3}{2}f_{k}-2f_{k-1}+\frac{1}{2}f_{k-2}
\right]+ O[(\Delta x)^{2}]$  




DST
(Differentiated Stirling):
$\displaystyle {F_{k}}'$ $\textstyle \approx$ $\displaystyle \frac{1}{\Delta x} \left[ \mu \delta \mbox{$f_{k}$}- \frac{1}{6} ...
...{3} \mbox{$f_{k}$}
+ \frac{1}{30} \mu \delta^{5} \mbox{$f_{k}$}
+ \dots \right]$  



Example:
$\displaystyle {F_{k}}'$ $\textstyle =$ $\displaystyle \frac{1}{\Delta x}\left[\mu \delta f_{k} \right]+ O[(\Delta x)^{2}]$  
  $\textstyle =$ $\displaystyle \frac{1}{2\Delta x} \left[f_{k+1}-f_{k-1}
\right]+ O[(\Delta x)^{2}]$  





Figure: Comparison of various simple approximations to the first differential quotient:


$\displaystyle DNGF:\;\; F_{k}'$ $\textstyle =$ $\displaystyle \frac{\Delta \mbox{$f_{k}$}}{\Delta x}
+ O[\Delta x]
=\frac{1}{\Delta x} \left[ f_{k+1}-f_{k} \right]
+O[\Delta x]$  
$\displaystyle DNGB:\;\; F_{k}'$ $\textstyle =$ $\displaystyle \frac{\nabla \mbox{$f_{k}$}}{\Delta x}
+ O[\Delta x]
= \frac{1}{\Delta x} \left[ f_{k}-f_{k-1} \right]
+ O[\Delta x]$  
$\displaystyle DST: \;\;\;F_{k}'$ $\textstyle =$ $\displaystyle \frac{\mu \delta \mbox{$f_{k}$}}{\Delta x}
+ O[(\Delta x)^{2}]
= \frac{1}{2\Delta x} \left[ f_{k+1}-f_{k-1} \right]
+ O[(\Delta x)^{2}]$  




next up previous
Next: 1.3.2 Second Derivatives Up: 1.3 Difference Quotients Previous: 1.3 Difference Quotients
Franz J. Vesely Oct 2005
See also:
"Computational Physics - An Introduction," Kluwer-Plenum 2001