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Jitse Niesen (talk | contribs) revert - one step of secant method improves solution by factor 1.6, so two steps improve it by 1.6^2 = 2.6 (approximately); feel free to clarify this if necessary |
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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_\star</math> that satisifies <math>f(x_\star)=0</math>. Near the solution <math>x_\star</math>, the function <math>f</math> is supposed to approximately satisfy <math>-1 < f'(x_\star) < 0</math>, which makes it adequate as an correction function for finding its own solution, although it is not required to be efficient. 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.
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. 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>[[Germund Dahlquist]], Åke Björck, tr. Ned Anderson (1974) ''Numerical Methods'', pp. 230-231, Prentice Hall, Englewood Cliffs, NJ</ref>:
:<math>x_{n+1} = x_n - \frac{f(x_n)}{g(x_n)}</math>
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