The idea is to use quadratic interpolation to approximate the inverse of f. This algorithm is rarely used on its own, but it is important because it forms part of the popular Brent's method.
The inverse quadratic interpolation algorithm is defined by the recurrence relation where fk = f(xk).
The asymptotic behaviour is very good: generally, the iterates xn converge fast to the root once they get close.
For instance, if by any chance two of the function values fn−2, fn−1 and fn coincide, the algorithm fails completely.
As noted in the introduction, inverse quadratic interpolation is used in Brent's method.