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  "body": "\n## Statement\n\nLet $p$ be a polynomial of degree $n \\ge 2$ with complex coefficients.\nFor a point $x$ with $p(x) \\ne 0$, put\n$$G = \\frac{p'(x)}{p(x)}, \\qquad H = G^2 - \\frac{p''(x)}{p(x)}, \\qquad a = \\frac{n}{G \\pm \\sqrt{(n - 1)(nH - G^2)}},$$\nwhere the sign is chosen so that the denominator has the larger absolute value.\n*Laguerre's method* is the iteration $x_{k+1} = x_k - a$, with $G, H, a$ computed at $x = x_k$.\nThen:\n\n1. If $r$ is a [[root|019MA1VTR2]] of $p$ of [[multiplicity|047SX3TD69]] $1$, then for $x_0$ close enough to $r$ the iterates [[converge|YWZQC5E54B]] to $r$ with order $3$.\n2. If all roots of $p$ are real, then for every real $x_0$ the iterates stay real and converge to a root of $p$.\n\n## Remarks\n\nThe formula comes from a model of $p$ near $x$.\nAssume that one root $r$ is at distance $a = x - r$, and that the other $n - 1$ roots all sit at a common distance $b$.\nThe identities $G = \\sum_i \\frac{1}{x - r_i}$ and $H = \\sum_i \\frac{1}{(x - r_i)^2}$ over the roots $r_i$ then give two equations for $a$ and $b$, whose solution is the formula above.\nFor polynomials with only real roots, a real starting point between two consecutive roots leads to one of these two roots.\nA real starting point outside all roots leads to the nearest root on that side.\nAt a multiple root the convergence is only linear.\nThe square root can be non-real, so the method also finds the [[complex|09FTAET5Z4]] roots of real polynomials.\n\n## Explanations\n\n### L1 human\n\nLaguerre's method is a root finder made for polynomials.\nIt pretends that one root is near and all the others are bunched together far away, and it solves this simplified picture exactly.\nThe answer is a very good next guess, and near a simple root the number of correct digits roughly triples at each step.\n\n### L2 human\n\nThe method uses $p$, $p'$ and $p''$, which Horner's scheme evaluates cheaply.\nIts convergence from almost any start makes it a robust choice for finding all roots one at a time, with deflation after each root.\nIts behaviour on polynomials with only real roots is a rare case of guaranteed global convergence.\n",
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