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  "body": "\n## Summary\n\nNewton's method finds approximate zeros of a differentiable function $f$.\nFrom a starting point $x_0$, it computes\n$$x_{n+1} = x_n - \\frac{f(x_n)}{f'(x_n)}.$$\nEach new point is the zero of the [[tangent line|01R1HW3Y42]] to the graph at the old point.\n\n## Remarks\n\nAssume $f$ is $C^2$ near a zero $r$ with $f'(r) \\ne 0$. Then the method converges to $r$ for $x_0$ close to $r$. The convergence is quadratic: $|x_{n+1} - r| \\le C |x_n - r|^2$.\nFor $f(x) = x^2 - 2$, the step is $x_{n+1} = (x_n + 2/x_n)/2$. This is Heron's method for $\\sqrt{2}$.\nThe method can fail. It stops when $f'(x_n) = 0$, and it can cycle or diverge from a poor starting point.\n\n## Explanations\n\n### L1 human\n\nReplace the graph by its tangent line, and find where the line crosses the axis.\nRepeat from there. Near a simple root, each step roughly doubles the number of correct digits.\n",
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