Editing Finite difference (section)

==Higher-order differences==
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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'' + {{sfrac|''h''|2}})}}  and {{math|''f''′(''x'' − {{sfrac|''h''|2}})}}   and applying a central difference formula for the derivative of {{math|''f''′}} at {{mvar|x}}, we obtain the central difference approximation of the second derivative of {{mvar|f}}:

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

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

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

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

;Forward :<math>\Delta^n_h[f](x) = \sum_{i = 0}^{n} (-1)^{n-i} \binom{n}{i} f\bigl(x + i h\bigr),</math>
;Backward :<math>\nabla^n_h[f](x) = \sum_{i = 0}^{n} (-1)^i \binom{n}{i} f(x - ih),</math>
;Central :<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).</math>

These equations use [[binomial coefficient]]s after the summation sign shown as {{math|<big><big>(</big></big>{{su|p=''n''|b=''i''|a=c}}<big><big>)</big></big>}}.  Each row of [[Pascal's triangle]] provides the coefficient for each value of {{mvar|i}}.

Note that the central difference will, for odd {{mvar|n}}, have {{mvar|h}}  multiplied by non-integers. This is often a problem because it amounts to changing the interval of discretization. The problem may be remedied substituting the average of  <math>\ \delta^n[f](\ x - \tfrac{\ h\ }{ 2 }\ )\ </math>  and  <math>\ \delta^n[f](\ x + \tfrac{\ h\ }{ 2 }\ ) ~.</math>

Forward differences applied to a [[sequence]] are sometimes called the [[binomial transform]] of the sequence, and have a number of interesting combinatorial properties. Forward differences may be evaluated using the [[Nörlund&ndash;Rice integral]]. The integral representation for these types of series is interesting, because the integral can often be evaluated using [[asymptotic expansion]] or [[saddle-point]] techniques; by contrast, the forward difference series can be extremely hard to evaluate numerically, because the binomial coefficients grow rapidly for large {{mvar|n}}.

The relationship of these higher-order differences with the respective derivatives is  straightforward,
<math display="block">\frac{d^n f}{d x^n}(x) = \frac{\Delta_h^n[f](x)}{h^n}+o(h) = \frac{\nabla_h^n[f](x)}{h^n}+o(h) = \frac{\delta_h^n[f](x)}{h^n} + o\left(h^2\right).</math>

Higher-order differences can also be used to construct better approximations. As mentioned above, the first-order difference approximates the first-order derivative up to a term of order {{mvar|h}}. However, the combination
<math display="block"> \frac{\Delta_h[f](x) - \frac12 \Delta_h^2[f](x)}{h} = - \frac{f(x+2h)-4f(x+h)+3f(x)}{2h} </math>
approximates {{math|''f''′(''x'')}} up to a term of order {{math|''h''<sup>2</sup>}}. This can be proven by expanding the above expression in [[Taylor series]], or by using the calculus of finite differences, explained below.

If necessary, the finite difference can be centered about any point by mixing forward, backward, and central differences.