Jenkins–Traub algorithm: Difference between revisions

Revision as of 15:43, 6 March 2008

Revision as of 19:34, 4 April 2008

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Given a polynomial ''P'',

::::<math>P(z)=\sum_{i=0}^na_iz^{n-i}, \quad a_0=1,\quad a_n\ne 0</math>

with complex coefficients compute approximations to the ''n'' zeros <math>\alpha_1,\alpha_1,\dots,\alpha_n</math> of ''P''(''z'').

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The <math>H^{(\lambda)}(z)</math> are now generated using the variable shifts <math>s_{\lambda}</math> which are generated by

::::<math>s_{\lambda+1}=s_\lambda- \frac{P(s_\lambda)}{\bar H^{(\lambda+1)}(s_\lambda)}, \quad \lambda=L,L+1,\dots,</math>

where <math>\bar H^{(\lambda+1)}(z)</math> is <math>H^{(\lambda)}(z)</math> divided by its leading coefficient.