Jenkins–Traub algorithm: Difference between revisions

Revision as of 23:49, 26 September 2023

Revision as of 12:20, 8 November 2023

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To that end, a sequence of so-called ''H'' polynomials is constructed. These polynomials are all of degree ''n''&nbsp;&minus;&nbsp;1 and are supposed to converge to the factor <math>\bar H(X)</math> of ''P''(''X'') containing (the linear factors of) all the remaining roots. The sequence of ''H'' polynomials occurs in two variants, an unnormalized variant that allows easy theoretical insights and a normalized variant of <math>\bar H</math> polynomials that keeps the coefficients in a numerically sensible range.

The construction of the ''H'' polynomials <math>\left(H^{(\lambda)}(z)\right)_{\lambda=0,1,2,\dots}</math> is guided by a sequence of [[complex numbersnumber]]s <math>(s_\lambda)_{\lambda=0,1,2,\dots}</math> called shifts. These shifts themselves depend, at least in the third stage, on the previous ''H'' polynomials. The ''H'' polynomials are defined as the solution to the implicit recursion

<math display="block">

H^{(0)}(z)=P^\prime(z)