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  "body": "\n## Statement\n\nLet $f$ be a real function that is twice continuously differentiable on an open interval $I$.\nLet $r \\in I$ be a simple zero of $f$, that is, $f(r) = 0$ and $f'(r) \\ne 0$.\nFor $x$ with $f(x) \\ne 0$, put\n$$g(x) = \\frac{f(x + f(x)) - f(x)}{f(x)}.$$\n*Steffensen's method* is the iteration\n$$x_{n+1} = x_n - \\frac{f(x_n)}{g(x_n)},$$\nstopped when $f(x_n) = 0$.\nThen there is $\\delta \u003e 0$ such that for every $x_0$ with $\\abs{x_0 - r} \u003c \\delta$ the iterates are defined and [[converge|YWZQC5E54B]] to $r$ quadratically.\n\n## Remarks\n\nThe quantity $g(x)$ is a difference quotient of $f$ with step $h = f(x)$.\nNear $r$ the step $f(x)$ is small, and $g(x) = f'(x) + O(f(x))$.\nSo the method behaves like [[Newton's method|07EXVNF9VP]], with an error in the slope of the same size as the current error, which keeps the order $2$.\nEach step costs two evaluations of $f$ and no derivative.\nThe step is exactly [[Aitken's delta-squared process|WF08BMN6Q6]] applied to the points $x$, $\\varphi(x)$, $\\varphi(\\varphi(x))$ of the fixed-point map $\\varphi(x) = x + f(x)$.\nThe method needs $x_0$ close to $r$: when $f(x_0)$ is large, the point $x_0 + f(x_0)$ can be far away and the slope estimate is useless.\n\n## Explanations\n\n### L1 human\n\nNewton's method needs the derivative.\nSteffensen's method replaces it by the slope between $x$ and $x + f(x)$.\nNear the root $f(x)$ is tiny, so this slope is very accurate, and the method is as fast as Newton's method.\n\n### L2 human\n\nIt is the derivative-free counterpart of Newton's method, with the same quadratic order.\nIt also shows how convergence acceleration turns a linearly convergent fixed-point iteration into a quadratically convergent one.\nIts basin of convergence is often smaller than that of Newton's method.\n",
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