Lady Windermere's Fan (mathematics)

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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[edit]

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_{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[edit]

\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_{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{align}

See also[edit]