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 [math]\displaystyle{ E(\ \tau,t_0,y(t_0)\ ) }[/math] be the exact solution operator so that:
- [math]\displaystyle{ y(t_0+\tau) = E(\tau,t_0,y(t_0))\ y(t_0) }[/math]
with [math]\displaystyle{ t_0 }[/math] denoting the initial time and [math]\displaystyle{ y(t) }[/math] the function to be approximated with a given [math]\displaystyle{ y(t_0) }[/math].
Further let [math]\displaystyle{ y_n }[/math], [math]\displaystyle{ n \in \N,\ n\le N }[/math] be the numerical approximation at time [math]\displaystyle{ t_n }[/math], [math]\displaystyle{ t_0 \lt t_n \le T = t_N }[/math]. [math]\displaystyle{ y_n }[/math] can be attained by means of the approximation operator [math]\displaystyle{ \Phi(\ h_n,t_n,y(t_n)\ ) }[/math] so that:
- [math]\displaystyle{ y_n = \Phi(\ h_{n-1},t_{n-1},y(t_{n-1})\ )\ y_{n-1}\quad }[/math] with [math]\displaystyle{ h_n = t_{n+1} - t_n }[/math]
The approximation operator represents the numerical scheme used. For a simple explicit forward Euler method with step width [math]\displaystyle{ h }[/math] this would be: [math]\displaystyle{ \Phi_{\text{Euler}}(\ h,t_{n-1},y(t_{n-1})\ )\ y_{n-1} = (1 + h \frac{d}{dt})\ y_{n-1} }[/math]
The local error [math]\displaystyle{ d_n }[/math] is then given by:
- [math]\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} }[/math]
In abbreviation we write:
- [math]\displaystyle{ \Phi(h_n) := \Phi(\ h_n,t_n,y(t_n)\ ) }[/math]
- [math]\displaystyle{ E(h_n) := E(\ h_n,t_n,y(t_n)\ ) }[/math]
- [math]\displaystyle{ D(h_n) := D(\ h_n,t_n,y(t_n)\ ) }[/math]
Then Lady Windermere's Fan for a function of a single variable [math]\displaystyle{ t }[/math] writes as:
[math]\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 }[/math]
with a global error of [math]\displaystyle{ y_N-y(t_N) }[/math]
Explanation
[math]\displaystyle{ \begin{align} 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{align} }[/math]
See also
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