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 play Lady Windermere's Fan, A Play About a Good Woman.

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 $t_{n}$ , $t_{0}<t_{n}\leq T=t_{N}$ . $y_{n}$ 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 witdth $h$ 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

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

Oscar Wilde's Lady Windermere's Fan
Films	Lady Windermere's Fan (1916) Lady Windermere's Fan (1925) El abanico de Lady Windermere (1944) The Fan (1949) A Good Woman (2004)
Other	After the Ball (musical) Lady Windermere syndrome Lady Windermere's Fan (mathematics)

Lady Windermere's Fan (mathematics)

Lady Windermere's Fan for a function of one variable

Explanation

See also