数值分析中,理查德森外推法(Richardson extrapolation)用以改善级数序列收敛效率,它是在20世纪前期由英国数学家,物理学家,气象学家Lewis Fry Richardson提出的。在数值分析领域,Richardson外推法有很多实际应用,如Romberg's method,是在梯形公式的基础上应用Richardson外推法导出的;还有用于求解常微分方程的Bulirsch–Stoer算法。 推导 假定某一函数 D {\displaystyle D} 可数值近似(离散化)为 D ( h ) {\displaystyle D\left(h\right)} ,其中 h {\displaystyle h} 为步长, D = D ( h ) + a h p + b h q + … {\displaystyle D=D\left(h\right)+ah^{p}+bh^{q}+\ldots } (1) 其中 p {\displaystyle p} 为首项阶数, q {\displaystyle q} 下一项阶数, 满足 q > p {\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 h 2 {\displaystyle h_{2}} 做离散近似 D = D ( h 2 ) + a h 2 p + b h 2 q + … {\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 h p {\displaystyle h^{p}} 项,我们可以对以上两式相减,即(1) − r {\displaystyle -r} (2),其中 r = ( h h 2 ) p {\displaystyle r=\left({\frac {h}{h_{2}}}\right)^{p}} : ( 1 − r ) D = D ( h ) + a h p + b h q − r D ( h 2 ) − r a h 2 p − r b h 2 q + … = D ( h ) − r D ( h 2 ) + a ( h p − r h 2 p ) ⏟ 0 + b ( h q − r h 2 q ) {\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)} ( 1 − r ) D = D ( h ) − r D ( h 2 ) + b ( h q − r h 2 q ) {\displaystyle \left(1-r\right)D=D\left(h\right)-rD\left(h_{2}\right)+b\left(h^{q}-rh_{2}^{q}\right)} D = D ( h ) − r D ( h 2 ) 1 − r + b ( h q − r h 2 q ) 1 − r = D ( h ) − r D ( h 2 ) 1 − r + b ( 1 − r h 2 q h q ) h q 1 − r {\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}}} D = D ( h ) − r D ( h 2 ) 1 − r + b ( 1 − ( h h 2 ) p ( h 2 h ) q ) h q 1 − r = D ( h ) − r D ( h 2 ) 1 − r ⏟ D ∗ ( h ) + b ( 1 − ( h 2 h ) q − p ) 1 − r ⏟ b ∗ h q {\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: ∴ D = D ∗ ( h ) + b ∗ h q {\displaystyle \therefore D=D^{\ast }\left(h\right)+b^{\ast }h^{q}} D ∗ ( h ) {\displaystyle D^{\ast }(h)} 代替了 D ( h ) {\displaystyle D(h)} ,为 D {\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。 示例 应用理查德森方法,改善用于近似微分的中心差分公式 f ′ ( x n ) = f ( x n + h ) − f ( x n − h ) 2 h ⏟ D ( h ) − f ‴ ( x n ) 6 ⏟ a h 2 − f ( 5 ) ( x n ) 120 ⏟ b h 4 {\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 p = 2 , q = 4 {\displaystyle p=2,q=4} , h 2 = 2 h , r = ( 1 2 ) p = 1 4 {\displaystyle h_{2}=2h,r=\left({\frac {1}{2}}\right)^{p}={\frac {1}{4}}} 代入公式: D ∗ = D ( h ) − r D ( h 2 ) 1 − r = f ( x n + h ) − f ( x n − h ) 2 h − 1 4 f ( x n + 2 h ) − f ( x n − 2 h ) 4 h 1 − 1 4 {\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}}}}} D ∗ = 8 [ f ( x n + h ) − f ( x n − h ) ] − f ( x n + 2 h ) + f ( x n − 2 h ) 12 h {\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。 参考文献 Extrapolation Methods. Theory and Practice by C. Brezinski and M. Redivo Zaglia, North-Holland, 1991. 外部链接 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) Loading related searches...Wikiwand - on Seamless Wikipedia browsing. On steroids.