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[edit]
Let
be the exact solution operator so that:
![{\displaystyle y(t_{0}+\tau )=E(\tau ,t_{0},y(t_{0}))\ y(t_{0})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e1475cac668c255df812152a6f5420ae61faa335)
with
denoting the initial time and
the function to be approximated with a given
.
Further let
,
be the numerical approximation at time
,
.
can be attained by means of the approximation operator
so that:
with ![{\displaystyle h_{n}=t_{n+1}-t_{n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b0ed5683e5de2a265521251ade066685b57cfb30)
The approximation operator represents the numerical scheme used. For a simple explicit forward Euler method with step width
this would be:
The local error
is then given by:
![{\displaystyle 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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a95750df0760038dfd29743092dbf6de1154c7e4)
In abbreviation we write:
![{\displaystyle \Phi (h_{n}):=\Phi (\ h_{n},t_{n},y(t_{n})\ )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ec207f2d34abe8409baa0f74693e03779f02fcfd)
![{\displaystyle E(h_{n}):=E(\ h_{n},t_{n},y(t_{n})\ )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c4ad7ffb893b5573d3b22b0275a810c92d83ef39)
![{\displaystyle D(h_{n}):=D(\ h_{n},t_{n},y(t_{n})\ )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/66ed9522fba630706870fa6e86c8ece15a571608)
Then Lady Windermere's Fan for a function of a single variable
writes as:
with a global error of
Explanation[edit]
See also[edit]