Jenkins–Traub algorithm: Difference between revisions

Revision as of 10:16, 3 August 2009

Revision as of 10:18, 3 August 2009

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s_{\lambda+1}

=s_\lambda- \frac{P(s_\lambda)}{\bar H^{\lambda+1}(s_\lambda)}

=s_\lambda-\frac{W^\lambda(s_\lambda)}{(W^\lambda)'(s_\lambda)}
</math>

is precisely a Newton-Raphson iteration performed on certain [[rational functions]]. More precisely, Newton-Raphson is being performed on a sequence of rational functions

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For <math>\lambda</math> sufficiently large,

:<math>\frac{P(z)/}{\bar H^{(\lambda}(z)}=W^\lambda(z)\,LC(H^{\lambda})</math>

is as close as desired to a first degree polynomial