Lady Windermere's Fan (mathematics) explained

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 _{n-1} ,t _{n-1} ,y(t _{n-1} ) ) y _{n-1} with 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 this would be: \Phi _{Euler} ( h,t _{n-1} ,y(t _{n-1} ) ) y _{n-1} = ( 1 + h (d)/(dt) ) y _{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 ^{N -1} _{j =0} \Phi(h _{j) (y} _{0-y(t} _{0))} + \sum ^{N -1} _{j = n} \Phi(h _{j) d} _{n}

with a global error of

y _{N-y(t} _{N)}

Explanation

\begin{align} y _{N} - y(t _{N)} &{} = y _{N} - \underbrace{\prod ^{N -1} _{j =0} \Phi(h _{j) y(t} _{0)} + \prod ^{N -1} _{j =0} \Phi(h _{j) y(t} _{0)}} _{=0} - y(t _{N)} \\ &{} = y _{N} - \prod ^{N -1} _{j =0} \Phi(h _{j) y(t} _{0)} + \underbrace{\sum ^{N -1} _{n =0} \prod ^{N -1} _{j = n} \Phi(h _{j) y(t} _{n)} - \sum ^{N -1} _{j = n} \Phi(h _{j) y(t} _{n)}} ^{= \prod ^{N -1} _{j =0} \Phi(h _{j) y(t} _{0)-\sum} ^{N} ^{n = N} \left[\prod ^{N -1} _{j = n} \Phi(h _{j)\right] y(t} _{n)} = \prod ^{N -1} _{j =0} \Phi(h _{j) y(t} _{0)} - y(t _{N)}} \\ &{} = \prod ^{N -1} _{j =0} \Phi(h _{j) y} _{0} - \prod ^{N -1} _{j =0} \Phi(h _{j) y(t} _{0)} + \sum ^{N -1} _{j = n -1} \Phi(h _{j) y(t} _{n-1} ) - \sum ^{N -1} _{j = n} \Phi(h _{j) y(t} _{n)} \\ &{} = \prod ^{N -1} _{j =0} \Phi(h _{j) (y} _{0-y(t} _{0))} + \sum ^{N -1} _{j = n} \Phi(h _{j)} \left[ \Phi(h _{n-1} ) - E(h _{n-1} ) \right] y(t _{n-1} ) \\ &{} = \prod ^{N -1} _{j =0} \Phi(h _{j) (y} _{0-y(t} _{0))} + \sum ^{N -1} _{j = n} \Phi(h _{j) d} _{n \end{align}}

See also

• Baker–Campbell–Hausdorff formula
• Numerical error