Steffensen's method: Difference between revisions

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{{short description|Newton-like root-finding algorithm that does not use derivatives}}

In [[numerical analysis]], '''Steffensen's method''' is an [[iterative method]] for numerical [[root-finding method|root-finding]] named after [[Johan Frederik Steffensen]] that is similar to the [[secant method]] and to [[Newton's method]]. '''Steffensen's method''' achieves similara quadratic [[order of convergence|quadratic convergence]] without using [[derivative]]s, whereas [[Newton's method]] converges quadratically but requires derivatives and the secant method does not require derivatives but also converges less quickly than quadratically. Steffenson's method has the drawback that it requires two function evaluations per step, however, whereas Newton's method and the secant method require only one evaluation per step, so it is not necessarily most efficient in terms of [[computational cost]].

Steffenson's method can be derived as an adaptation of [[Aitken's delta-squared process]] applied to [[fixed-point iteration]]. Viewed in this way, Steffenson's method naturally generalizes to efficient fixed-point calculation in general [[Banach space|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]].

==Simple description==