有限差分係數

中心差分估算一阶至高阶微分按照下式：

${\displaystyle f^{(n)}(x)={\frac {1}{h_{x}^{n}}}\sum _{i=-m}^{m}k_{i}f(x+ih)+O(h_{x}^{p})}$

其中${\displaystyle h_{x}}$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。[1]

例如，${\displaystyle h_{x}^{2}}$精度的三阶导的中心差分式为

${\displaystyle \displaystyle f'''(x_{0})\approx \displaystyle {\frac {-{\frac {1}{2}}f(x_{-2})+f(x_{-1})-f(x_{+1})+{\frac {1}{2}}f(x_{+2})}{h_{x}^{3}}}+O\left(h_{x}^{2}\right)}$

下表列出不同精度下，等间距的一阶至高阶前向差分系数。[1]

例如，${\displaystyle h_{x}^{3}}$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

${\displaystyle \displaystyle f'(x_{0})\approx \displaystyle {\frac {-{\frac {11}{6}}f(x_{0})+3f(x_{+1})-{\frac {3}{2}}f(x_{+2})+{\frac {1}{3}}f(x_{+3})}{h_{x}}}+O\left(h_{x}^{3}\right),}$

${\displaystyle h_{x}^{2}}$精度二阶导的前向差分式为

${\displaystyle \displaystyle f''(x_{0})\approx \displaystyle {\frac {2f(x_{0})-5f(x_{+1})+4f(x_{+2})-f(x_{+3})}{h_{x}^{2}}}+O\left(h_{x}^{2}\right),}$

对应的后向差分式分别为

${\displaystyle \displaystyle f'(x_{0})\approx \displaystyle {\frac {{\frac {11}{6}}f(x_{0})-3f(x_{-1})+{\frac {3}{2}}f(x_{-2})-{\frac {1}{3}}f(x_{-3})}{h_{x}}}+O\left(h_{x}^{3}\right),}$

${\displaystyle \displaystyle f''(x_{0})\approx \displaystyle {\frac {2f(x_{0})-5f(x_{-1})+4f(x_{-2})-f(x_{-3})}{h_{x}^{2}}}+O\left(h_{x}^{2}\right),}$

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：

• 有限差分法
• 數值微分