Successive parabolic interpolation


Successive parabolic interpolation is a technique for finding the extremum of a continuous unimodal function by successively fitting parabolas to a function of one variable at three unique points or, in general, a function of n variables at 1+n/2 points, and at each iteration replacing the "oldest" point with the extremum of the fitted parabola.

Advantages

Only function values are used, and when this method converges to an extremum, it does so with an order of convergence of approximately 1.325. The superlinear rate of convergence is superior to that of other methods with only linear convergence. Moreover, not requiring the computation or approximation of function derivatives makes successive parabolic interpolation a popular alternative to other methods that do require them.

Disadvantages

On the other hand, convergence is not guaranteed when using this method in isolation. For example, if the three points are collinear, the resulting parabola is degenerate and thus does not provide a new candidate point. Furthermore, if function derivatives are available, Newton's method is applicable and exhibits quadratic convergence.

Improvements

Alternating the parabolic iterations with a more robust method to choose candidates can greatly increase the probability of convergence without hampering the convergence rate.