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→As a multidimensional Newton's method: Added section about the interpretation as an inverse power iteration. |
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\end{align}</math>
From this formula one recovers <math>s_{k+1}=s_k-\tfrac{P(s_k)}{\bar H^k(s_k)}</math> and
that <math>\bar H^{k+1}</math> is again a normed polynomial, since ''P'' has leading coefficient ''1'' and <math>\bar H^k</math> has one degree less, not influencing the leading coefficient of the quotient. Note that the Jenkins-Traub algorithm uses a further improved shift update in that there
:<math>s_{k+1}=s_k-\tfrac{P(s_k)}{\bar H^{k+1}(s_k)}</math>.
In the first and second stage, the shifts <math>s_k</math> and consequently the first polynomial <math>G^k</math> are left constant, that is without update. The ensuing method is best understood as an inverse power iteration.
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