|
The inverse quadratic interpolation algorithm is defined by the recurrence relation
where fk = f(xk). As can be seen from the recurrence relation, this method requires three initial values, x0, x1 and x2.
We use the three preceding iterates, xn−2, xn−1 and xn, with their function values, fn−2, fn−1 and fn. Applying the Lagrange interpolation formula to do quadratic interpolation on the inverse of f yields
We are looking for a root of f, so we substitute y = f(x) = 0 in the above equation and this results in the above recursion formula.
The asymptotic behaviour is very good: generally, the iterates xn converge fast to the root once they get close. However, performance is often quite poor if you do not start very close to the actual root. For instance, if by any chance two of the function values fn−2, fn−1 and fn coincide, the algorithm fails completely. Thus, inverse quadratic interpolation is seldom used as a stand-alone algorithm.
The order of this convergence is approximately 1.8, it can be proved by the Secant Method analysis.
As noted in the introduction, inverse quadratic interpolation is used in Brent's method.
Inverse quadratic interpolation is also closely related to some other root-finding methods. Using linear interpolation instead of quadratic interpolation gives the secant method. Interpolating f instead of the inverse of f gives Müller's method.
ssearcha Business c Business tsearchW