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{{NumBlk|:|<math> x_{n+k+1} = x_{n+k} - \frac{f(x_{n+k})}{f'(x_{n+k})} </math>|{{EquationRef|2}}}}
This is the [[Newton's method|Newton-Raphson method]]. It starts off with a single estimate <math>x_1</math> so we can take ''k'' = 0 in {{EquationNote|2}}. It does not require an interpolating polynomial but instead one has to
Once the interpolating polynomial <math>p_{n,k} (x)</math> has been calculated, one can also calculate the next estimate <math>x_{n+k+1}</math> as a solution of <math>p_{n,k} (x)=0</math> instead of using {{EquationNote|1}}. For ''k'' = 1 these two methods are identical: it is the [[secant method]]. For ''k'' = 2 this method is known as [[Muller's method]]. For ''k'' = 3 this approach involves finding the roots of [[cubic function]], which is unattractively complicated. This problem aggravates for even larger values of ''k''. An additional complication is that the equation <math>p_{n,k} (x)=0</math> will in general have [[Properties of polynomial roots|multiple solutions]] and a prescription has to be given which of these solutions is the next estimate <math>x_{n+k+1}</math>. Muller does this for the case ''k'' = 2 but no such prescriptions appear to exist for k > 2.
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