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  "body": "\n## Summary\n\nFixed-point iteration computes a solution of $f(x) = x$ by applying $f$ again and again to a starting point.\nIf the resulting sequence converges and $f$ is continuous, its limit is a fixed point of $f$.\nThe [[Banach fixed point theorem|0VK6VEH345]] guarantees convergence for a contraction on a complete metric space.\nMany numerical methods, such as [[Newton's method|07EXVNF9VP]], are fixed-point iterations.\n\n## Explanations\n\n### L1 human\n\nPick a number, apply a function to it, apply the function to the result, and keep going.\nIf the numbers settle down, they settle at a point that the function does not move.\nTry it with the cosine key of a calculator in radians: starting from $1$, the values approach about $0.739085$, the only real solution of $\\cos x = x$.\nIteration is a simple way to solve an equation when one can rewrite it as $x = f(x)$.\n",
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