Steffensen's method

In numerical analysis, Steffensen's method is an iterative method for numerical root-finding named after Johan Frederik Steffensen that is similar to the secant method and to Newton's method. Steffensen's method achieves a quadratic order of convergence without using derivatives, whereas the more familiar Newton's method also converges quadratically, but requires derivatives and the secant method does not require derivatives but also converges less quickly than quadratically.

Steffensen's method has the drawback that it requires two function evaluations per step, whereas the secant method requires only one evaluation per step, so it is not necessarily most efficient in terms of computational cost, depending on the number of iterations each requires. Newton's method also requires evaluating two functions per step – for the function and for its derivative – and its computational cost varies between being the same as Steffensen's method (for most functions, where calculation of the derivative is just as computationally costly as the original function).^[a]

Steffensen's method can be derived as an adaptation of Aitken's delta-squared process applied to fixed-point iteration. Viewed in this way, Steffensen's method naturally generalizes to efficient fixed-point calculation in general Banach spaces, whenever fixed points are guaranteed to exist and fixed-point iteration is guaranteed to converge, although possibly slowly, by the Banach fixed-point theorem.
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