Steffensen's method: Difference between revisions

Revision as of 03:00, 3 August 2008

Revision as of 03:11, 3 August 2008

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:<math>g(x_n) = \frac{f(x_n + f(x_n)) - f(x_n)}{f(x_n)}</math>

The function <math>g</math> is the average slope of the function &fnof; between the last sequence point <math>(x,y)=( x_n,\ f(x_n))\ )</math> and the auxilliary point <math>(x,y)=( x_n + f(x_n)h,\ f(x_n + h) )</math>, with the step <math>h=f(x_n)))\ </math>&nbsp;. It is only for the purpose of finding this auxilliary point that the function <math>f</math> must be an adequate itteration function for its own solution, to get quick convergence for the sequence of values <math>x_n</math>&nbsp;. For all other parts of the calculation, Steffensen's method only requires the function <math>f</math> to be continuous, and to actually have a solution. Several modest modifications of the slope function <math>g</math> exist to accomodate functions <math>f</math> that do not meet this requirement.

The main advantage of Steffensen's method is that it can find the roots of an equation <math>f</math> just as "[[quadratic convergence|quickly]]" as [[Newton's method]] but the formula does not require a separate function for the derivative, so it can be programmed for any generic function. In this case ''[[quadratic convergence|quicly]]'' means that the number of correct digits in the answer doubles with each step. The cost for the quick convergence is the double function evaluation: both <math>f(x_n)</math> and <math>f(x_n + f(x_n))</math> must be calculated, which might be time-consuming if <math>f</math> is a complicated function.