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| In [[mathematics]], '''Lady Windermere's Fan''' is a telescopic identity employed to relate global and local error of a [[Numerical analysis|numerical algorithm]]. The name is derived from [[Oscar Wilde]]'s play [[Lady Windermere's Fan|Lady Windermere's Fan, A Play About a Good Woman]].
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| ==Lady Windermere's Fan for a function of one variable==
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| Let <math>E(\ \tau,t_0,y(t_0)\ )</math> be the '''exact solution operator''' so that:
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| ::<math>y(t_0+\tau) = E(\tau,t_0,y(t_0))\ y(t_0)</math>
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| with <math>t_0</math> denoting the initial time and <math>y(t)</math> the function to be approximated with a given <math>y(t_0)</math>.
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| Further let <math>y_n</math>, <math>n \in \N,\ n\le N</math> be the numerical approximation at time <math>t_n</math>, <math>t_0 < t_n \le T = t_N</math>. <math>y_n</math> can be attained by means of the '''approximation operator''' <math>\Phi(\ h_n,t_n,y(t_n)\ )</math> so that:
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| ::<math>y_n = \Phi(\ h_{n-1},t_{n-1},y(t_{n-1})\ )\ y_{n-1}\quad</math> with <math>h_n = t_{n+1} - t_n</math>
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| The approximation operator represents the numerical scheme used. For a simple explicit forward [[Euler method|euler scheme]] with step witdth <math>h</math> this would be: <math>\Phi_{\text{Euler}}(\ h,t_{n-1},y(t_{n-1})\ )\ y(t_{n-1}) = (1 + h \frac{d}{dt})\ y(t_{n-1})</math>
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| The '''local error''' <math>d_n</math> is then given by:
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| ::<math>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>
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| In abbreviation we write:
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| ::<math>\Phi(h_n) := \Phi(\ h_n,t_n,y(t_n)\ )</math>
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| ::<math>E(h_n) := E(\ h_n,t_n,y(t_n)\ )</math>
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| ::<math>D(h_n) := D(\ h_n,t_n,y(t_n)\ )</math>
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| Then '''Lady Windermere's Fan''' for a function of a single variable <math>t</math> writes as:
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| <math>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>
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| with a global error of <math>y_N-y(t_N)</math>
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| ===Explanation===
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| <math>\begin{align}
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| y_N - y(t_N) &{}=
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| 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) \\
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| &{}= 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) } \\
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| &{}= \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) \\
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| &{}= \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}) \\
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| &{}= \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
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| \end{align}</math>
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| ==See also==
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| * [[Baker–Campbell–Hausdorff formula]]
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| * [[Numerical error]]
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| {{Lady Windermere's Fan}}
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| {{DEFAULTSORT:Lady Windermere's Fan (Mathematics)}}
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| [[Category:Numerical analysis]]
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My name is Millie (23 years old) and my hobbies are Exhibition Drill and People watching.
Also visit my blog post ... quality retailing