Fixed-Point Iteration

Updated 01 Oct 2026

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What it is
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  rewrites a root equation as $x=g(x)$ and repeatedly evaluates $x_{k+1}=g(x_k)$
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Convergence
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  a sufficient interval condition is that $g$ maps a closed interval into itself and satisfies $|g'(x)|\le q<1$ throughout that interval.
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  locally, $|g'(x^*)|<1$ indicates attraction; $|g'(x^*)|>1$ indicates repulsion. Equality to $1$ is inconclusive.
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Example
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  for $f(x)=x^2-3x+2$ , one rearrangement is $g(x)=-2/(x-3)$ , with $x\ne3$ .
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  since $g'(x)=2/(x-3)^2$ , root $1$ is attractive and root $2$ is repelling.
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  another rearrangement is $g(x)=(x^2+2)/3$ . Here $g'(1)=2/3$ and $g'(2)=4/3$ : again, locally attractive at $1$ and repelling at $2$ .
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Notes
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  The fraction needs the negative numerator: $x=-2/(x-3)$ .
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  The second rearrangement is not universally divergent; convergence depends on the root, starting point, and iteration behavior.

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