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Concept
Steffensen's method
Résumé
In numerical analysis, Steffensen's method is an iterative method for root-finding named after Johan Frederik Steffensen which is similar to Newton's method, but with certain situational advantages. In particular, Steffensen's method achieves similar quadratic convergence, but without using derivatives as Newton's method does.
The simplest form of the formula for Steffensen's method occurs when it is used to find a zero of a real function f; that is, to find the real value that satisfies Near the solution the function is supposed to approximately satisfy this condition makes adequate as a correction-function for for finding its own solution, although it is not required to work efficiently. For some functions, Steffensen's method can work even if this condition is not met, but in such a case, the starting value must be very close to the actual solution and convergence to the solution may be slow.
Given an adequate starting value a sequence of values can be generated using the formula below. When it works, each value in the sequence is much closer to the solution than the prior value. The value from the current step generates the value for the next step, via this formula:
for where the slope function is a composite of the original function given by the following formula:
or perhaps more clearly,
where is a step-size between the last iteration point, x, and an auxiliary point located at
The function is the average value for the slope of the function between the last sequence point and the auxiliary point with the step It is also called the first-order divided difference of between those two points.
It is only for the purpose of finding for this auxiliary point that the value of the function must be an adequate correction to get closer to its own solution, and for that reason fulfill the requirement that For all other parts of the calculation, Steffensen's method only requires the function to be continuous and to actually have a nearby solution.
Source officielle
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