Inverse quadratic interpolation

Inverse quadratic interpolation is a root-finding algorithm of numerical analysis, used to solve equations of the form f(x) = 0. Instead of interpolating the function itself, it fits a quadratic through three earlier points by treating x as a function of y, that is, by interpolating the inverse of f with the Lagrange interpolation formula, and then reads off the value at y = 0 as the next estimate of the root. Each step needs the three preceding iterates and their function values, so the method has to be started from three initial values. Once the iterates are close to the root it converges quickly, with an order of about 1.84, but it often performs poorly when the starting values are far from the root, and it fails outright if two of the function values coincide. For that reason it is seldom used on its own; its main importance is as one component of Brent's method. Using linear interpolation in its place gives the secant method, and interpolating f rather than its inverse gives Muller's method.