Lady Windermere's Fan (mathematics)

Template:One source 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 $E(\tau ,t_0,y(t_0))$ 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 $y_n$, $n\in \mathbb{N},n\le N$ be the numerical approximation at time $t_n$, $t_0<t_n\le T=t_N$. $y_n$ can be attained by means of the approximation operator $\mathrm{\Phi }(h_n,t_n,y(t_n))$ so that:

$y_n=\mathrm{\Phi }(h_{n-1},t_{n-1},y(t_{n-1}))y_{n-1}$ with $h_n=t_{n+1}-t_n$

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

The local error $d_n$ is then given by:

$d_n:=D(h_{n-1},t_{n-1},y(t_{n-1}))y_{n-1}:=[\mathrm{\Phi }(h_{n-1},t_{n-1},y(t_{n-1}))-E(h_{n-1},t_{n-1},y(t_{n-1}))]y_{n-1}$

In abbreviation we write:

$\mathrm{\Phi }(h_n):=\mathrm{\Phi }(h_n,t_n,y(t_n))$

$E(h_n):=E(h_n,t_n,y(t_n))$

$D(h_n):=D(h_n,t_n,y(t_n))$

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

$y_N-y(t_N)=\prod _{j=0}^{N-1}\mathrm{\Phi }(h_j)(y_0-y(t_0))+\sum _{n=1}^N\prod _{j=n}^{N-1}\mathrm{\Phi }(h_j)d_n$

with a global error of $y_N-y(t_N)$

Explanation

$\begin{array}{cc}{y}_N-y(t_N)& =y_N-\underset{=0}{\underbrace{\prod _{j=0}^{N-1}\mathrm{\Phi }(h_j)y(t_0)+\prod _{j=0}^{N-1}\mathrm{\Phi }(h_j)y(t_0)}}-y(t_N)\\ & =y_N-\prod _{j=0}^{N-1}\mathrm{\Phi }(h_j)y(t_0)+\underset{=\prod _{j=0}^{N-1}\mathrm{\Phi }(h_j)y(t_0)-\sum _{n=N}^N[\prod _{j=n}^{N-1}\mathrm{\Phi }(h_j)]y(t_n)=\prod _{j=0}^{N-1}\mathrm{\Phi }(h_j)y(t_0)-y(t_N)}{\underbrace{\sum _{n=0}^{N-1}\prod _{j=n}^{N-1}\mathrm{\Phi }(h_j)y(t_n)-\sum _{n=1}^N\prod _{j=n}^{N-1}\mathrm{\Phi }(h_j)y(t_n)}}\\ & =\prod _{j=0}^{N-1}\mathrm{\Phi }(h_j)y_0-\prod _{j=0}^{N-1}\mathrm{\Phi }(h_j)y(t_0)+\sum _{n=1}^N\prod _{j=n-1}^{N-1}\mathrm{\Phi }(h_j)y(t_{n-1})-\sum _{n=1}^N\prod _{j=n}^{N-1}\mathrm{\Phi }(h_j)y(t_n)\\ & =\prod _{j=0}^{N-1}\mathrm{\Phi }(h_j)(y_0-y(t_0))+\sum _{n=1}^N\prod _{j=n}^{N-1}\mathrm{\Phi }(h_j)[\mathrm{\Phi }(h_{n-1})-E(h_{n-1})]y(t_{n-1})\\ & =\prod _{j=0}^{N-1}\mathrm{\Phi }(h_j)(y_0-y(t_0))+\sum _{n=1}^N\prod _{j=n}^{N-1}\mathrm{\Phi }(h_j)d_n\end{array}$

See also

• Baker–Campbell–Hausdorff formula
• Numerical error