Question

Write Stirling's central difference formula.​

Answer 1

Answer:

Stirling's formula decrease much more rapidly than other difference formulae hence considering first few number of terms itself will give better accuracy. Forward or backward difference formulae use the oneside information of the function where as Stirling's formula uses the function values on both sides of f(x).

Answer 2

Stirling's central difference formula is $f_{1} - 2f_{0} + f_{-1}$.

Given

To write the stirling's central difference formula  

Consider a function f( x ), which have been equally spaced between the values $x_{0} , x_{1} , x_{2}, x_{3} , ............x_{n}$ with h.

Where, "h" is the step length.

$\delta |$f( x ) = f ( x + $\frac{h}{2}$ ) - f ( x - $\frac{h}{2}$ )    

$\delta |$$f_{i}$ = ( $E^{\frac{1}{2} } - E^{\frac{- 1}{2} }$ ) $f_{i}$

     = ( $f_{i + \frac{1}{2} } - f_{i - \frac{1}{2} }$ )

$\delta |^{2}$$f_{i}$ = ( $E^{\frac{1}{2} }$ - $E^{\frac{- 1}{2} }$ ) ( $f_{i + \frac{1}{2} } - f_{i - \frac{1}{2} }$ )

       = $f_{1} - f_{0} - f_{0} + f_{-1}$

       = $f_{1} - 2f_{0} + f_{-1}$

Therefore, stirling's central difference is $f_{1} - 2f_{0} + f_{-1}$.

To learn more...

Write Stirling's central difference formula.​

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