Lady Windermere's Fan (mathematics)

In mathematics, Lady Windermere's Fan is a telescopic identity employed to relate global and local error of a numerical algorithm. The name is derived from Oscar Wilde's 1892 play Lady Windermere's Fan, A Play About a Good Woman .

Lady Windermere's Fan for a function of one variable

Let ${\displaystyle E(\ \tau ,t_{0},y(t_{0})\ )}$ be the exact solution operator so that:

${\displaystyle y(t_{0}+\tau )=E(\tau ,t_{0},y(t_{0}))\ y(t_{0})}$

with ${\displaystyle t_{0}}$ denoting the initial time and ${\displaystyle y(t)}$ the function to be approximated with a given ${\displaystyle y(t_{0})}$.

Further let ${\displaystyle y_{n}}$, ${\displaystyle n\in \mathbb {N} ,\ n\leq N}$ be the numerical approximation at time ${\displaystyle t_{n}}$, ${\displaystyle t_{0}<t_{n}\leq T=t_{N}}$. ${\displaystyle y_{n}}$ can be attained by means of the approximation operator${\displaystyle \Phi (\ h_{n},t_{n},y(t_{n})\ )}$ so that:

${\displaystyle y_{n}=\Phi (\ h_{n-1},t_{n-1},y(t_{n-1})\ )\ y_{n-1}\quad }$ with ${\displaystyle h_{n}=t_{n+1}-t_{n}}$

The approximation operator represents the numerical scheme used. For a simple explicit forward Euler method with step width ${\displaystyle h}$ this would be: ${\displaystyle \Phi _{\text{Euler}}(\ h,t_{n-1},y(t_{n-1})\ )\ y_{n-1}=(1+h{\frac {d}{dt}})\ y_{n-1}}$

The local error${\displaystyle d_{n}}$ is then given by:

${\displaystyle d_{n}:=D(\ h_{n-1},t_{n-1},y(t_{n-1})\ )\ y_{n-1}:=\left[\Phi (\ h_{n-1},t_{n-1},y(t_{n-1})\ )-E(\ h_{n-1},t_{n-1},y(t_{n-1})\ )\right]\ y_{n-1}}$

In abbreviation we write:

${\displaystyle \Phi (h_{n}):=\Phi (\ h_{n},t_{n},y(t_{n})\ )}$

${\displaystyle E(h_{n}):=E(\ h_{n},t_{n},y(t_{n})\ )}$

${\displaystyle D(h_{n}):=D(\ h_{n},t_{n},y(t_{n})\ )}$

Then Lady Windermere's Fan for a function of a single variable ${\displaystyle t}$ writes as:

${\displaystyle y_{N}-y(t_{N})=\prod _{j=0}^{N-1}\Phi (h_{j})\ (y_{0}-y(t_{0}))+\sum _{n=1}^{N}\ \prod _{j=n}^{N-1}\Phi (h_{j})\ d_{n}}$

with a global error of ${\displaystyle y_{N}-y(t_{N})}$

Explanation

${\displaystyle {\begin{aligned}y_{N}-y(t_{N})&{}=y_{N}-\underbrace {\prod _{j=0}^{N-1}\Phi (h_{j})\ y(t_{0})+\prod _{j=0}^{N-1}\Phi (h_{j})\ y(t_{0})} _{=0}-y(t_{N})\\&{}=y_{N}-\prod _{j=0}^{N-1}\Phi (h_{j})\ y(t_{0})+\underbrace {\sum _{n=0}^{N-1}\ \prod _{j=n}^{N-1}\Phi (h_{j})\ y(t_{n})-\sum _{n=1}^{N}\ \prod _{j=n}^{N-1}\Phi (h_{j})\ y(t_{n})} _{=\prod _{j=0}^{N-1}\Phi (h_{j})\ y(t_{0})-\sum _{n=N}^{N}\left[\prod _{j=n}^{N-1}\Phi (h_{j})\right]\ y(t_{n})=\prod _{j=0}^{N-1}\Phi (h_{j})\ y(t_{0})-y(t_{N})}\\&{}=\prod _{j=0}^{N-1}\Phi (h_{j})\ y_{0}-\prod _{j=0}^{N-1}\Phi (h_{j})\ y(t_{0})+\sum _{n=1}^{N}\ \prod _{j=n-1}^{N-1}\Phi (h_{j})\ y(t_{n-1})-\sum _{n=1}^{N}\ \prod _{j=n}^{N-1}\Phi (h_{j})\ y(t_{n})\\&{}=\prod _{j=0}^{N-1}\Phi (h_{j})\ (y_{0}-y(t_{0}))+\sum _{n=1}^{N}\ \prod _{j=n}^{N-1}\Phi (h_{j})\left[\Phi (h_{n-1})-E(h_{n-1})\right]\ y(t_{n-1})\\&{}=\prod _{j=0}^{N-1}\Phi (h_{j})\ (y_{0}-y(t_{0}))+\sum _{n=1}^{N}\ \prod _{j=n}^{N-1}\Phi (h_{j})\ d_{n}\end{aligned}}}$

See also

• Baker–Campbell–Hausdorff formula
• Numerical error