Fixed-point iteration

In numerical analysis, fixed-point iteration is a method of computing fixed points of iterated functions.

More specifically, given a function ${\displaystyle f}$ defined on the real numbers with real values and given a point ${\displaystyle x_{0}}$ in the domain of ${\displaystyle f}$, the fixed point iteration is

which gives rise to the sequence ${\displaystyle x_{0},x_{1},x_{2},\dots }$ which is hoped to converge to a point ${\displaystyle x}$. If ${\displaystyle f}$ is continuous, then one can prove that the obtained ${\displaystyle x}$ is a fixed point of ${\displaystyle f}$, i.e.,

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