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  "body": "\n## Statement\n\nLet $f$ be a real function that is three times continuously differentiable on an open interval $I$, and let $r \\in I$ satisfy $f(r) = 0$ and $f'(r) \\ne 0$.\nGiven three points $x_{n-2}, x_{n-1}, x_n$ whose values $f_k = f(x_k)$ are pairwise distinct, *inverse quadratic interpolation* defines\n$$x_{n+1} = \\frac{f_{n-1} f_n \\, x_{n-2}}{(f_{n-2} - f_{n-1})(f_{n-2} - f_n)} + \\frac{f_{n-2} f_n \\, x_{n-1}}{(f_{n-1} - f_{n-2})(f_{n-1} - f_n)} + \\frac{f_{n-2} f_{n-1} \\, x_n}{(f_n - f_{n-2})(f_n - f_{n-1})}.$$\nThen there is $\\delta \u003e 0$ with the following property.\nFor all starting points $x_0, x_1, x_2$ 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 new point is the value at $y = 0$ of the [[Lagrange interpolation|237KR6FRCK]] polynomial of degree at most $2$ through the points $(f_k, x_k)$.\nThis polynomial approximates the [[inverse function|0J2WAR83WP]] of $f$ near $r$, so its value at $0$ approximates $r = f^{-1}(0)$.\nThe error analysis gives $e_{n+1} \\approx C e_n e_{n-1} e_{n-2}$ for the errors $e_k = x_k - r$, which leads to the stated order.\nInterpolating $x$ as a function of $y$ avoids solving a quadratic equation, unlike Muller's method.\nThe method breaks down when two of the values $f_k$ coincide, and it can diverge from poor starting points.\nFor this reason it is used inside safeguarded methods such as Brent's method.\nWith two points instead of three, the same idea gives the [[secant method|9ZC177PVPC]].\n\n## Explanations\n\n### L1 human\n\nSwap the roles of $x$ and $y$: think of $x$ as a function of $y = f(x)$.\nFit a parabola in this swapped picture through three known points, and read off where it meets $y = 0$.\nThat reading is the next guess for the root.\n\n### L2 human\n\nThe method needs no derivatives and only one new function value per step.\nIts order is a little less than $2$ but better than the secant method's $1.618$.\nIt is fast but fragile, so libraries combine it with bisection to keep a bracket around the root.\n",
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