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Milds 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.

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## Lady Windermere's Fan for a function of one variable

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

$y(t_{0}+\tau )=E(\tau ,t_{0},y(t_{0}))\ y(t_{0})$ with $t_{0}$ denoting the initial time and $y(t)$ the function to be approximated with a given $y(t_{0})$ .

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

$y_{n}=\Phi (\ h_{n-1},t_{n-1},y(t_{n-1})\ )\ y_{n-1}\quad$ with $h_{n}=t_{n+1}-t_{n}$ The approximation operator represents the numerical scheme used. For a simple explicit forward euler scheme with step width $h$ this would be: $\Phi _{\text{Euler}}(\ h,t_{n-1},y(t_{n-1})\ )\ y(t_{n-1})=(1+h{\frac {d}{dt}})\ y(t_{n-1})$ The local error $d_{n}$ is then given by:

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

$\Phi (h_{n}):=\Phi (\ h_{n},t_{n},y(t_{n})\ )$ $E(h_{n}):=E(\ h_{n},t_{n},y(t_{n})\ )$ $D(h_{n}):=D(\ h_{n},t_{n},y(t_{n})\ )$ Then Lady Windermere's Fan for a function of a single variable $t$ writes as:

$y_{N}-y(t_{N})=\prod _{j=0}^{N-1}\Phi (h_{j})\ (y_{0}-y(t_{0}))+\sum _{n=1}^{N}\ \prod _{j=n}^{N-1}\Phi (h_{j})\ d_{n}$ with a global error of $y_{N}-y(t_{N})$ ### Explanation

{\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 _{n=0}^{N-1}\Phi (h_{n})\ y(t_{n})-\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}}  This page was last edited on 26 April 2021, at 20:58