Parabolic interpolation

Given a triplet $(a,b,c)$ that brackets a minimum, we approximate the objective function in the interval $(a,c)$ with the parabola fitting the triplet. Then we find the minimum of this parabola with the formula (since we want the abscissa, the method is indeed an inverse parabolic interpolation):

$$x^\star=b-\frac{1}{2}\frac{(b-a)^2[f(b)-f(c)]-(b-c)^2[f(b)-f(a)]}{(b-a)[f(b)-f(c)]-(b-c)[f(b)-f(a)]}$$

This method is useful only when the function is quite smooth in the interval, but it has the advantage that the convergence is almost quadratic, and it is perfectly quadratic when the function to be optimized is a quadratic form.