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  "body": "\n## Statement\n\nLet $f$ be a complex function that is holomorphic near a simple zero $r$, so $f(r) = 0$ and $f'(r) \\ne 0$.\nGiven three distinct points $x_{k-2}, x_{k-1}, x_k$, write $f[x_i, x_j] = \\dfrac{f(x_i) - f(x_j)}{x_i - x_j}$ and $f[x_k, x_{k-1}, x_{k-2}] = \\dfrac{f[x_k, x_{k-1}] - f[x_{k-1}, x_{k-2}]}{x_k - x_{k-2}}$, and put\n$$w = f[x_k, x_{k-1}] + f[x_k, x_{k-2}] - f[x_{k-1}, x_{k-2}].$$\n*Muller's method* defines\n$$x_{k+1} = x_k - \\frac{2 f(x_k)}{w \\pm \\sqrt{w^2 - 4 f(x_k) \\, f[x_k, x_{k-1}, x_{k-2}]}},$$\nwhere the sign is chosen so that the denominator has the larger absolute value.\nThen there is $\\delta \u003e 0$ with the following property.\nFor all starting points within $\\delta$ of $r$ for which the iteration is defined, the iterates [[converge|YWZQC5E54B]] to $r$ with order $q \\approx 1.839$.\nHere $q$ is the real root of $q^3 = q^2 + q + 1$.\n\n## Remarks\n\nThe point $x_{k+1}$ is the zero, closest to $x_k$, of the [[interpolating polynomial|SS1YMS2NPC]] of degree at most $2$ through $(x_{k-2}, f(x_{k-2}))$, $(x_{k-1}, f(x_{k-1}))$ and $(x_k, f(x_k))$.\nThe formula is the quadratic formula written in the stable form that avoids cancellation.\nThe errors $e_k = x_k - r$ satisfy $e_{k+1} \\approx C e_k e_{k-1} e_{k-2}$, which gives the stated order.\nBecause of the square root, the iterates can leave the real line, so the method finds [[complex|09FTAET5Z4]] zeros even from real starting points.\nThis makes it useful for zeros of real polynomials that are not real.\nWith a line through two points instead of a parabola through three, the same idea gives the [[secant method|9ZC177PVPC]].\n\n## Explanations\n\n### L1 human\n\nThe secant method draws a line through two points of the graph and takes its crossing with the axis.\nMuller's method draws a parabola through three points and takes its crossing nearest to the last point.\nA parabola can miss the real axis, and then the method moves into the complex numbers, which lets it find complex roots.\n\n### L2 human\n\nThe method needs no derivatives and one new function value per step.\nIts order is about $1.84$, between the secant method and Newton's method.\nIt is a common choice for finding all roots of a polynomial one at a time, combined with deflation.\n",
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