Төгсгөлөг ялгавар: Засвар хоорондын ялгаа

In an analogous way one can obtain finite difference approximations to higher order derivatives and differential operators. For example, by using the above central difference formula for {{math|''f ' ''(''x''+''h''/2)}} and {{math|''f ' ''(''x''−''h''/2)}} and applying a central difference formula for the derivative of {{mvar| f '}} at {{mvar|x}}, we obtain the central difference approximation of the second derivative of {{mvar|f}}:

 

:<math> f''(x) \approx \frac{\delta_h^2[f](x)}{h^2} = \frac{f(x+h) - 2 f(x) + f(x-h)}{h^{2}} . </math>

 

Similarly we can apply other differencing formulas in a recursive manner.

 

:<math> f''(x) \approx \frac{\Delta_h^2[f](x)}{h^2} = \frac{f(x+2h) - 2 f(x+h) + f(x)}{h^{2}} . </math>

 

More generally, the '''{{mvar|n}}-th order forward, backward, and central''' differences are given by, respectively,

 

:<math>\Delta^n_h[f](x) =

\sum_{i = 0}^{n} (-1)^i \binom{n}{i} f(x + (n - i) h),

:<math>\Delta^n [f](x)= \sum_{k=0}^n\binom nk(-1)^{n-k}f(x + k)</math>

 

:<math>\nabla^n_h[f](x) =

\sum_{i = 0}^{n} (-1)^i \binom{n}{i} f(x - ih),

</math>

 

:<math>\delta^n_h[f](x) =

\sum_{i = 0}^{n} (-1)^i \binom{n}{i} f\left(x + \left(\frac{n}{2} - i\right) h\right).