What is Fixed point iteration: Definition and 19 Discussions

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.

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