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 the zeros, or roots, of a function <math>f\ </math>, that is, to find the input value <math>x\ </math> that satisifies <math>f(x)=0\ </math>. Given an adequate starting value <math>x_0\ </math>, a sequence of values <math>x_0,\ x_1,\ x_2,\ ...,\ x_n ...</math> can be generated. EachWhen it works, each value in the sequence is much closer to the solution than the prior value,. and theThe value <math>x_n\ </math> from the priorcurrent step generates the next step,value <math>x_{n+1}\ </math> for the next step, via this formula<ref>Germund Dahlquist, Åke Björck, tr. Ned Anderson (1974) ''Numerical Methods'', pp.&nbsp;230-231, Prentice Hall, Englewood Cliffs, NJ</ref>:

:<math>x_{n+1} = x_n - \frac{f(x_n)}{g(x_n)}</math>