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Fixed point iteration
In numerical analysis, fixed-point iteration is a method of computing fixed points of a function.
More specifically, given a function
f
{\displaystyle f}
defined on the real numbers with real values and given a point
x
0
{\displaystyle x_{0}}
in the domain of
f
{\displaystyle f}
, the fixed point iteration is
x
n
+
1
=
f
(
x
n
)
,
n
=
0
,
1
,
2
,
…
{\displaystyle x_{n+1}=f(x_{n}),\,n=0,1,2,\dots }
which gives rise to the sequence
x
0
,
x
1
,
x
2
,
…
{\displaystyle x_{0},x_{1},x_{2},\dots }
which is hoped to converge to a point
x
{\displaystyle x}
. If
f
{\displaystyle f}
is continuous, then one can prove that the obtained
x
{\displaystyle x}
is a fixed point of
f
{\displaystyle f}
, i.e.,
f
(
x
)
=
x
.
{\displaystyle f(x)=x.\,}
More generally, the function
f
{\displaystyle f}
can be defined on any metric space with values in that same space.
View More On Wikipedia.org
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