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下表列出使用[[有限差分法]]进行数值微分时，各项的系数。按计算中自变量取值方向，分为'''中心差分'''，'''前向差分'''和'''后向差分'''。 ==中心差分== 中心差分估算一阶至高阶微分按照下式： :<math>f^{(n)}(x) = \frac{1}{h_x^n}\sum_{i=-m}^{m} k_{i} f(x+ih) + O(h_x^p)</math> 其中<math> h_x </math>为自变量取等距格点计算函数值时的间隔。 下表列出不同计算精度下，等间距的一阶至高阶中心差分系数。<ref name=fornberg>{{Citation | last1=Fornberg | first1=Bengt | title=Generation of Finite Difference Formulas on Arbitrarily Spaced Grids | doi=10.1090/S0025-5718-1988-0935077-0 | year=1988 | journal={{tsl|en|Mathematics of Computation||Mathematics of Computation}} | issn=0025-5718 | volume=51 | issue=184 | pages=699–706}}.</ref> {| class="wikitable" style="text-align:center" |- ! 阶次 ! 精度 ! −4 ! −3 ! −2 ! −1 ! 0 ! 1 ! 2 ! 3 ! 4 |- | rowspan="4" | 1 || 2 || || || || −1/2 || 0|| 1/2|| || || |- || 4 || || || 1/12 || −2/3 || 0|| 2/3|| −1/12 || || |- || 6 || || −1/60 || 3/20 || −3/4 || 0 || 3/4 || −3/20 || 1/60 || |- style="border-bottom: 2px solid #aaa;" || 8 ||1/280 || −4/105 || 1/5 || −4/5 || 0 || 4/5 || −1/5 || 4/105 || −1/280 |- | rowspan="4" | 2 || 2 || || || || 1 || −2|| 1|| || || |- || 4 || || || −1/12 || 4/3 || −5/2|| 4/3|| −1/12 || || |- || 6 || || 1/90 || −3/20 || 3/2 || −49/18 || 3/2 || −3/20 || 1/90 || |- style="border-bottom: 2px solid #aaa;" || 8 ||−1/560 || 8/315 || −1/5 || 8/5 || −205/72 || 8/5 || −1/5 || 8/315 || −1/560 |- | rowspan="3" | 3 || 2 || || || −1/2 || 1 || 0|| −1|| 1/2 || || |- || 4 || || 1/8 || −1 || 13/8 || 0|| −13/8|| 1 || −1/8 || |- style="border-bottom: 2px solid #aaa;" || 6 || −7/240 || 3/10 || −169/120 || 61/30 ||0 || −61/30|| 169/120 || −3/10 || 7/240 |- | rowspan="3" | 4 || 2 || || || 1 || −4 || 6|| −4|| 1 || || |- || 4 || || −1/6 || 2 || −13/2 || 28/3|| −13/2|| 2 || −1/6 || |- style="border-bottom: 2px solid #aaa;" || 6 || 7/240 || −2/5 || 169/60 || −122/15 ||91/8 || −122/15|| 169/60 || −2/5 || 7/240 |- style="border-bottom: 2px solid #aaa;" | rowspan="1" | 5 || 2 || || −1/2 || 2 || −5/2 || 0|| 5/2|| −2 || 1/2 || |- | rowspan="1" | 6 || 2 || || 1 || −6 || 15 || −20 || 15 || −6 || 1 || |} 例如，<math>h_x^2</math>精度的三阶导的中心差分式为 : <math>\displaystyle f'''(x_{0}) \approx \displaystyle \frac{-\frac{1}{2}f(x_{-2}) + f(x_{-1}) -f(x_{+1}) +\frac{1}{2}f(x_{+2})}{h^3_x} + O\left(h_x^2 \right) </math> ==前向与后向差分== 下表列出不同精度下，等间距的一阶至高阶前向差分系数。<ref name=fornberg/> {| class="wikitable" style="text-align:center" |- ! 阶次 ! 精度 ! 0 ! 1 ! 2 ! 3 ! 4 ! 5 ! 6 ! 7 ! 8 |- | rowspan="6" | 1 || 1 || −1 || 1 || || || || || || || |- || 2 || −3/2 || 2 || −1/2 || || || || || || |- || 3 || −11/6 || 3 || −3/2|| 1/3 || || || || || |- || 4 || −25/12 || 4 || −3 || 4/3 || −1/4|| || || || |- || 5 || −137/60 || 5 || −5 || 10/3 || −5/4 || 1/5 || || || |- style="border-bottom: 2px solid #aaa;" || 6 || −49/20 || 6 || −15/2 || 20/3 || −15/4 || 6/5 || −1/6 || || |- | rowspan="6" | 2 || 1 || 1 || −2 || 1 || || || || || || |- || 2 || 2 || −5 || 4 || −1 || || || || || |- || 3 || 35/12 || −26/3 || 19/2 || −14/3 || 11/12 || || || || |- || 4 || 15/4 || −77/6 || 107/6 || −13 || 61/12 || −5/6|| || || |- || 5 || 203/45 || −87/5 || 117/4 || −254/9 || 33/2 || −27/5 || 137/180 || || |- style="border-bottom: 2px solid #aaa;" || 6 || 469/90 || −223/10 || 879/20 || −949/18 || 41 || −201/10 || 1019/180 || −7/10 || |- | rowspan="6" | 3 || 1 || −1 || 3 || −3 || 1 || || || || || |- || 2 || −5/2 || 9 || −12 || 7 || −3/2|| || || || |- || 3 || −17/4 || 71/4 || −59/2 || 49/2 || −41/4 || 7/4 || || || |- || 4 || −49/8 || 29 || −461/8 || 62 || −307/8 || 13 || −15/8 || || |- || 5 || −967/120 || 638/15 || −3929/40 || 389/3 || −2545/24 || 268/5 || −1849/120 || 29/15 || |- style="border-bottom: 2px solid #aaa;" || 6 || −801/80 || 349/6 || −18353/120 || 2391/10 || −1457/6 || 4891/30 || −561/8 || 527/30 || −469/240 |- | rowspan="5" | 4 || 1 || 1 || −4 || 6 || −4 || 1|| || || || |- || 2 || 3 || −14 || 26 || −24 || 11 || −2 || || || |- || 3 || 35/6 || −31 || 137/2 || −242/3 || 107/2 || −19 || 17/6 || || |- || 4 || 28/3 || −111/2 || 142 || −1219/6 || 176 || −185/2 || 82/3 || −7/2 || |- style="border-bottom: 2px solid #aaa;" || 5 || 1069/80 || −1316/15 || 15289/60 || −2144/5 || 10993/24 || −4772/15 || 2803/20 || −536/15 || 967/240 |} 例如，<math>h_x^3</math>精度一阶导的前向差分式为 For example, the first derivative with a third-order accuracy and the second derivative with a second-order accuracy are : <math>\displaystyle f'(x_{0}) \approx \displaystyle \frac{-\frac{11}{6}f(x_{0}) + 3f(x_{+1}) -\frac{3}{2}f(x_{+2}) +\frac{1}{3}f(x_{+3}) }{h_{x}} + O\left(h_{x}^3 \right), </math> <math>h_x^2</math>精度二阶导的前向差分式为 : <math>\displaystyle f''(x_{0}) \approx \displaystyle \frac{2f(x_{0}) - 5f(x_{+1}) + 4f(x_{+2}) - f(x_{+3}) }{h_{x}^2} + O\left(h_{x}^2 \right), </math> 对应的后向差分式分别为 : <math>\displaystyle f'(x_{0}) \approx \displaystyle \frac{\frac{11}{6}f(x_{0}) - 3f(x_{-1}) +\frac{3}{2}f(x_{-2}) -\frac{1}{3}f(x_{-3}) }{h_{x}} + O\left(h_{x}^3 \right), </math> : <math>\displaystyle f''(x_{0}) \approx \displaystyle \frac{2f(x_{0}) - 5f(x_{-1}) + 4f(x_{-2}) - f(x_{-3}) }{h_{x}^2} + O\left(h_{x}^2 \right), </math> 实际上，奇数阶后向差分式相对前向差分，各系数q取相反数；而偶数阶的则不变。如下表： {| class="wikitable" style="text-align:center" |- ! 阶次 ! 精度 ! −8 ! −7 ! −6 ! −5 ! −4 ! −3 ! −2 ! −1 ! 0 |- | rowspan="2" | 1 || 1 || || || || || || || || −1 || 1 |- || 2 || || || || || || || 1/2 || −2 || 3/2 |- | rowspan="2" | 2 || 1 || || || || || || || 1 || −2 || 1 |- || 2 || || || || || || −1 || 4 || −5 || 2 |- | rowspan="2" | 3 || 1 || || || || || || −1 || 3 || −3 || 1 |- || 2|| || || || || 3/2 || −7 || 12 || −9 || 5/2 |- | rowspan="2" | 4 || 1 || || || || || 1 || −4 || 6 || −4 || 1 |- || 2 || || || || −2 || 11 || −24 || 26 || −14 || 3 |} ==参见== * [[有限差分法]] * [[數值微分]] ==参考资料== {{reflist}} [[Category:有限差分]] [[Category:数值微分方程]]
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