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  "body": "\n## Summary\n\nAn iterative method solves a problem by producing a sequence of approximate solutions, each computed from the previous ones, that converges to a solution.\nIt contrasts with a direct method, which gives the answer after a fixed finite number of steps.\nIterative methods are the standard tool for nonlinear equations and for very large sparse linear systems, where direct elimination costs too much.\n\n## Explanations\n\n### L1 human\n\nStart with a guess, improve it by a fixed rule, and repeat until the guess is good enough.\n[[Newton's method|07EXVNF9VP]] for $\\sqrt{2}$ is an example: from $x_0 = 1$ the rule $x_{k+1} = (x_k + 2 / x_k) / 2$ gives $1.5$, then $1.41\\overline{6}$, then $1.41421\\ldots$.\nEach step is cheap, and one stops when the error is below a tolerance.\nThe questions are whether the sequence converges and how fast.\n",
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