Sidi's generalized secant method: Difference between revisions

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Latest revision as of 18:34, 22 March 2025

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{{NumBlk|:|<math> x_{n+k+1} = x_{n+k} - \frac{f(x_{n+k})}{p_{n,k}'(x_{n+k})}</math>|{{EquationRef|1}}}}

with <math>p_{n,k}'(x_{n+k})</math> the derivative of <math>p_{n,k}</math> at <math>x_{n+k}</math>. Having calculated <math>x_{n+k+1}</math> one calculates <math>f(x_{n+k+1})</math> and the algorithm can continue with the (''n''&nbsp;+&nbsp;1)th iteration. Clearly, this method requires the function <math>f</math> to be evaluated only once per iteration; it requires [[Derivative-free optimization|no derivatives]] of <math>f</math>.

The iterative cycle is stopped if an appropriate stop-stopping criterion is met. Typically the criterion is that the last calculated approximation is close enough to the sought-after root <math>\alpha</math>.

To execute the algorithm effectively, Sidi's method calculates the interpolating polynomial <math>p_{n,k} (x)</math> in its [[Newton polynomial|Newton form]].

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<references/>

{{root-finding algorithms}}

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[[Category:Root-finding algorithms]]