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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:
d
n
:=
D
(
h
n
−
1
,
t
n
−
1
,
y
(
t
n
−
1
)
)
y
n
−
1
:=
[
Φ
(
h
n
−
1
,
t
n
−
1
,
y
(
t
n
−
1
)
)
−
E
(
h
n
−
1
,
t
n
−
1
,
y
(
t
n
−
1
)
)
]
y
n
−
1
{\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}}
In abbreviation we write:
Φ
(
h
n
)
:=
Φ
(
h
n
,
t
n
,
y
(
t
n
)
)
{\displaystyle \Phi (h_{n}):=\Phi (\ h_{n},t_{n},y(t_{n})\ )}
E
(
h
n
)
:=
E
(
h
n
,
t
n
,
y
(
t
n
)
)
{\displaystyle E(h_{n}):=E(\ h_{n},t_{n},y(t_{n})\ )}
D
(
h
n
)
:=
D
(
h
n
,
t
n
,
y
(
t
n
)
)
{\displaystyle D(h_{n}):=D(\ h_{n},t_{n},y(t_{n})\ )}
Then Lady Windermere's Fan for a function of a single variable
t
{\displaystyle 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
{\displaystyle 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}}
with a global error of
y
N
−
y
(
t
N
)
{\displaystyle y_{N}-y(t_{N})}
Explanation [ ]
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
)
⏟
=
∏
j
=
0
N
−
1
Φ
(
h
j
)
y
(
t
0
)
−
∑
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
{\displaystyle {\begin{aligned}y_{N}-y(t_{N})&{}=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})\\&{}=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 _{j=0}^{N-1}\Phi (h_{j})\ y(t_{0})-\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})}\\&{}=\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})\\&{}=\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})\\&{}=\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}\end{aligned}}}
See also [ ]