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 $E(\ \tau,t_0,y(t_0)\ )$ 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 $y_n$, $n \in \N,\ n\le N$ 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 $\Phi(\ h_n,t_n,y(t_n)\ )$ so that:
::$y_n = \Phi(\ h_,t_,y(t_)\ )\ y_\quad$ with $h_n = t_ - t_n$

The approximation operator represents the numerical scheme used. For a simple explicit forward Euler method with step width $h$ this would be: $\Phi_(\ h,t_,y(t_)\ )\ y_ = (1 + h \frac)\ y_$

The local error $d_n$ is then given by:
::d_n:= D(\ h_,t_,y(t_)\ )\ y_ := \left \Phi(\ h_,t_,y(t_)\ ) - E(\ h_,t_,y(t_)\ ) \right y_

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:

$y_N-y(t_N) = \prod_^\Phi(h_j)\ (y_0-y(t_0)) + \sum_^N\ \prod_^ \Phi(h_j)\ d_n$

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

Explanation

\begin
y_N - y(t_N) &=
y_N - \underbrace_ - y(t_N) \\
&= y_N - \prod_^ \Phi(h_j)\ y(t_0) + \underbrace_ \\
&= \prod_^\Phi(h_j)\ y_0 - \prod_^\Phi(h_j)\ y(t_0) + \sum_^N\ \prod_^ \Phi(h_j)\ y(t_) - \sum_^N\ \prod_^ \Phi(h_j)\ y(t_n) \\
&= \prod_^\Phi(h_j)\ (y_0-y(t_0)) + \sum_^N\ \prod_^ \Phi(h_j) \left \Phi(h_) - E(h_) \right\ y(t_) \\
&= \prod_^\Phi(h_j)\ (y_0-y(t_0)) + \sum_^N\ \prod_^ \Phi(h_j)\ d_n
\end

See also

* Baker–Campbell–Hausdorff formula
* Numerical error

{{DEFAULTSORT:Lady Windermere's Fan (Mathematics)
Numerical analysis