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 ( τ , t 0 , y ( t 0 ) ) {\displaystyle E(\ \tau ,t_{0},y(t_{0})\ )} be the exact solution operator so that:
y ( t 0 + τ ) = E ( τ , t 0 , y ( t 0 ) ) y ( t 0 ) {\displaystyle y(t_{0}+\tau )=E(\tau ,t_{0},y(t_{0}))\ y(t_{0})}
with t 0 {\displaystyle t_{0}} denoting the initial time and y ( t ) {\displaystyle y(t)} the function to be approximated with a given y ( t 0 ) {\displaystyle y(t_{0})} . Further let y n {\displaystyle y_{n}} , n ∈ N , n ≤ N {\displaystyle n\in \mathbb {N} ,\ n\leq N} be the numerical approximation at time t n {\displaystyle t_{n}} , t 0 < t n ≤ T = t N {\displaystyle t_{0}<t_{n}\leq T=t_{N}} . y n {\displaystyle y_{n}} can be attained by means of the approximation operator Φ ( h n , t n , y ( t n ) ) {\displaystyle \Phi (\ h_{n},t_{n},y(t_{n})\ )} so that:
y n = Φ ( h n − 1 , t n − 1 , y ( t n − 1 ) ) y n − 1 {\displaystyle 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 {\displaystyle 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 {\displaystyle h} this would be: Φ Euler ( h , t n − 1 , y ( t n − 1 ) ) y n − 1 = ( 1 + h d d t ) y n − 1 {\displaystyle \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 {\displaystyle d_{n}} is then given by:
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