Steffensen's method

Steffensen's method is an iterative method of numerical root-finding named after Johan Frederik Steffensen, similar to the secant method and to Newton's method. It reaches quadratic order of convergence without using derivatives, whereas Newton's method also converges quadratically but needs derivatives, and the secant method needs none but converges more slowly than quadratically. Its drawback is that it needs two function evaluations per step against one for the secant method, so it is not automatically the cheapest, depending on how many iterations each method requires. Newton's method also needs two evaluations per step, one for the function and one for its derivative, and for most functions the derivative costs about as much as the function itself. Steffensen's method can be derived as an adaptation of Aitken's delta-squared process applied to fixed-point iteration, so it generalizes to fixed-point calculation in Banach spaces when fixed points exist and the iteration converges.