Sidi's generalized secant method: Difference between revisions

Revision as of 18:38, 10 April 2021

Revision as of 08:55, 1 July 2023

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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 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]].