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  "body": "\n## Summary\n\nA root-finding algorithm computes approximations of a zero of a function, that is, a point $x$ with $f(x) = 0$.\nMost such algorithms are iterative: they produce a sequence of points, or of shrinking intervals, that tends to a zero.\nBracketing methods such as the [[bisection method|7VMYC7MB3Q]] and [[regula falsi|462AEY7MWF]] keep a sign change and always converge for continuous $f$.\nOpen methods such as [[Newton's method|07EXVNF9VP]] and the [[secant method|9ZC177PVPC]] converge faster near a simple zero, but only from good starting points.\nMethods are compared by their [[rate of convergence|YWZQC5E54B]] and by the number of function evaluations per step.\n\n## Explanations\n\n### L1 human\n\nSolving an equation such as $x^3 = x + 1$ means finding where the function $f(x) = x^3 - x - 1$ is $0$.\nThere is no simple formula for most such equations, so one guesses and improves the guess step by step.\nFor this $f$, $f(1) = -1 \u003c 0$ and $f(2) = 5 \u003e 0$, so a zero lies between $1$ and $2$, and halving this interval repeatedly closes in on it, near $1.3247$.\nFaster methods use the slope of $f$ to jump close to the zero in a few steps.\n",
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