Steffensen's method: Difference between revisions

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==Simple description==

The simplest form of the formula for Steffensen's method occurs when it is used to find a [[zero of a function|zero]] of a [[real function]] <math>\ f\, ;</math> that is, to find the real value <math>x_\star</math> that satisfies <math>\ f(x_\star) = 0 ~.</math> Near the solution <math>\ x_\star\, ,</math> the derivative of the function, <math>\ f'\ ,</math> is supposed to approximately satisfy <math>\ -1 < f'(x_\star) < 0\, ;</math> this condition makes <math>\ f\ </math> adequate as a correction-function for <math>~ x ~</math> 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 <math>\ x_0\ </math> must be ''very'' close to the actual solution <math>\ x_\star\, ,</math> and convergence to the solution may be slow. Adjustment of the size of the method's intermediate step, mentioned later, can improve convergence in some of these cases.

Given an adequate starting value <math>\ x_0\, ,</math> a sequence of values <math>\ x_0,\ x_1,\ x_2, \dots,\ x_n,\ \dots\ </math> can be generated using the formula below. When it works, each value in the sequence is much closer to the solution <math>\ x_\star\ </math> than the prior value. The value <math>\ x_n\ </math> from the current step generates the value <math>\ x_{n+1}\ </math> for the next step, via this formula:<ref name=Dahlquist-Björck-1974/>