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.
Let E ( τ , t 0 , y ( t 0 ) ) be the exact solution operator so that:
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 ∈ N , n ≤ N be the numerical approximation at time t n , t 0 < t n ≤ T = t N . y n can be attained by means of the approximation operator Φ ( h n , t n , y ( t n ) ) so that:
The approximation operator represents the numerical scheme used. For a simple explicit forward euler scheme with step witdth h this would be: Φ Euler ( h , t n − 1 , y ( t n − 1 ) ) y ( t n − 1 ) = ( 1 + h d d t ) y ( t n − 1 )
The local error d n is then given by:
In abbreviation we write:
Then Lady Windermere's Fan for a function of a single variable t writes as:
y N − y ( t N ) = ∏ j = 0 N − 1 Φ ( h j ) ( y 0 − y ( t 0 ) ) + ∑ n = 1 N ∏ j = n N − 1 Φ ( h j ) d n
with a global error of y N − y ( t N )
y N − y ( t N ) = y N − ∏ j = 0 N − 1 Φ ( h j ) y ( t 0 ) + ∏ j = 0 N − 1 Φ ( h j ) y ( t 0 ) ⏟ = 0 − y ( t N ) = y N − ∏ j = 0 N − 1 Φ ( h j ) y ( t 0 ) + ∑ n = 0 N − 1 ∏ j = n N − 1 Φ ( h j ) y ( t n ) − ∑ n = 1 N ∏ j = n N − 1 Φ ( h j ) y ( t n ) ⏟ = ∏ n = 0 N − 1 Φ ( h n ) y ( t n ) − ∑ n = N N [ ∏ j = n N − 1 Φ ( h j ) ] y ( t n ) = ∏ j = 0 N − 1 Φ ( h j ) y ( t 0 ) − y ( t N ) = ∏ j = 0 N − 1 Φ ( h j ) y 0 − ∏ j = 0 N − 1 Φ ( h j ) y ( t 0 ) + ∑ n = 1 N ∏ j = n − 1 N − 1 Φ ( h j ) y ( t n − 1 ) − ∑ n = 1 N ∏ j = n N − 1 Φ ( h j ) y ( t n ) = ∏ j = 0 N − 1 Φ ( h j ) ( y 0 − y ( t 0 ) ) + ∑ n = 1 N ∏ j = n N − 1 Φ ( h j ) [ Φ ( h n − 1 ) − E ( h n − 1 ) ] y ( t n − 1 ) = ∏ j = 0 N − 1 Φ ( h j ) ( y 0 − y ( t 0 ) ) + ∑ n = 1 N ∏ j = n N − 1 Φ ( h j ) d n
