Steffensen's method: Difference between revisions

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In [[numerical analysis]], '''Steffensen's method''' is an [[iterative method]] named after [[Johan Frederik Steffensen]] for numerical [[root-finding method|root-finding]] that is similar to the [[secant method]] and to [[Newton's method]]. '''Steffensen's method''' achieves a quadratic [[order of convergence]] without using [[derivative]]s, 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 at best the same as the secant method, and at worst the same as Steffensen's method;. forFor most functions, where calculation of the derivative is just as computationally costly as the original function, so Newton's method is usually equally costly as Steffensen's.{{efn|

For rare special case functions the derivative for Newton's method can be calculated at negligible cost, by using saved parts from evaluation of the main function. If optimized in this way, Newton's method becomes only slightly more costly per step than the secant method, with slightly faster convergence.

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