理查德森外推法

假定某一函数${\displaystyle D}$ 可数值近似（离散化）为${\displaystyle D\left(h\right)}$ ，其中${\displaystyle h}$ 为步长，

${\displaystyle D=D\left(h\right)+ah^{p}+bh^{q}+\ldots }$  （1）

其中${\displaystyle p}$ 为首项阶数，${\displaystyle q}$ 下一项阶数, 满足${\displaystyle q>p}$ 。

考虑该函数又可以使用同样的数值近似方法，以步长为${\displaystyle h_{2}}$ 做离散近似

${\displaystyle D=D\left(h_{2}\right)+ah_{2}^{p}+bh_{2}^{q}+\ldots }$  （2）

如果希望消掉式(1)中的${\displaystyle h^{p}}$ 项，我们可以对以上两式相减，即(1)${\displaystyle -r}$ (2)，其中${\displaystyle r=\left({\frac {h}{h_{2}}}\right)^{p}}$ ：

${\displaystyle \left(1-r\right)D=D\left(h\right)+ah^{p}+bh^{q}-rD\left(h_{2}\right)-rah_{2}^{p}-rbh_{2}^{q}+\ldots =D\left(h\right)-rD\left(h_{2}\right)+a\underbrace {\left(h^{p}-rh_{2}^{p}\right)} _{0}+b\left(h^{q}-rh_{2}^{q}\right)}$ 

${\displaystyle \left(1-r\right)D=D\left(h\right)-rD\left(h_{2}\right)+b\left(h^{q}-rh_{2}^{q}\right)}$ 

${\displaystyle D={\frac {D\left(h\right)-rD\left(h_{2}\right)}{1-r}}+{\frac {b\left(h^{q}-rh_{2}^{q}\right)}{1-r}}={\frac {D\left(h\right)-rD\left(h_{2}\right)}{1-r}}+{\frac {b\left(1-r{\frac {h_{2}^{q}}{h^{q}}}\right)h^{q}}{1-r}}}$ 

${\displaystyle D={\frac {D\left(h\right)-rD\left(h_{2}\right)}{1-r}}+{\frac {b\left(1-\left({\frac {h}{h_{2}}}\right)^{p}\left({\frac {h_{2}}{h}}\right)^{q}\right)h^{q}}{1-r}}=\underbrace {\frac {D\left(h\right)-rD\left(h_{2}\right)}{1-r}} _{D^{\ast }\left(h\right)}+\underbrace {\frac {b\left(1-\left({\frac {h_{2}}{h}}\right)^{q-p}\right)}{1-r}} _{b^{\ast }}h^{q}}$ 

或简记作：

${\displaystyle \therefore D=D^{\ast }\left(h\right)+b^{\ast }h^{q}}$ 

${\displaystyle D^{\ast }(h)}$ 代替了${\displaystyle D(h)}$ ，为${\displaystyle D}$ 的新的数值近似。新近似相比最初形式具有更高阶的误差项，数值精度由此提高，此方法即为理查德森外推法。

应用理查德森方法，改善用于近似微分的中心差分公式

${\displaystyle f'\left(x_{n}\right)={\underset {D\left(h\right)}{\underbrace {\frac {f\left(x_{n}+h\right)-f\left(x_{n}-h\right)}{2h}} }}-{\underset {a}{\underbrace {\frac {f'''\left(x_{n}\right)}{6}} }}h^{2}-{\underset {b}{\underbrace {\frac {f^{\left(5\right)}\left(x_{n}\right)}{120}} }}h^{4}}$ 

则由式(1)可知${\displaystyle p=2,q=4}$ , ${\displaystyle h_{2}=2h,r=\left({\frac {1}{2}}\right)^{p}={\frac {1}{4}}}$  代入公式：

${\displaystyle D^{\ast }={\frac {D\left(h\right)-rD\left(h_{2}\right)}{1-r}}={\frac {{\frac {f\left(x_{n}+h\right)-f\left(x_{n}-h\right)}{2h}}-{\frac {1}{4}}{\frac {f\left(x_{n}+2h\right)-f\left(x_{n}-2h\right)}{4h}}}{1-{\frac {1}{4}}}}}$ 

${\displaystyle D^{\ast }={\frac {8\left[f\left(x_{n}+h\right)-f\left(x_{n}-h\right)\right]-f\left(x_{n}+2h\right)+f\left(x_{n}-2h\right)}{12h}}}$ 

由此，中心差分公式精度由2阶变为4阶。

• Extrapolation Methods. Theory and Practice by C. Brezinski and M. Redivo Zaglia, North-Holland, 1991.

• Module for Richardson's Extrapolation, fullerton.edu
• Fundamental Methods of Numerical Extrapolation With Applications （页面存档备份，存于互联网档案馆）, mit.edu
• Richardson-Extrapolation （页面存档备份，存于互联网档案馆）
• Richardson extrapolation on a website of Robert Israel (University of British Columbia) （页面存档备份，存于互联网档案馆）