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Fixed-point iteration

An equation $f(x) = 0$ can be rewritten as $x = g(x)$ and solved by iterating $x_{n+1} = g(x_n)$ from a starting value. The method only converges if $|g'(x)| < 1$ near the root — otherwise the iterations wander away from the answer instead of toward it.

Symbols

Example

$x = \cos x$ can be iterated as $x_{n+1} = \cos x_n$, which converges since $|\!-\!\sin x| < 1$ near the root.

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Part of Numerical Methods: Solving equations.