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Concept# Fixed-point iteration

Résumé

In numerical analysis, fixed-point iteration is a method of computing fixed points of a function.
More specifically, given a function f defined on the real numbers with real values and given a point x_0 in the domain of f, the fixed-point iteration is
x_{n+1}=f(x_n), , n=0, 1, 2, \dots
which gives rise to the sequence x_0, x_1, x_2, \dots of iterated function applications x_0, f(x_0), f(f(x_0)), \dots which is hoped to converge to a point x_\text{fix}. If f is continuous, then one can prove that the obtained x_\text{fix} is a fixed point of f, i.e.,
f(x_\text{fix})=x_\text{fix} .
More generally, the function f can be defined on any metric space with values in that same space.
Examples

- A first simple and useful example is the Babylonian method for computing the square root of a > 0, w

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