Lehmer–Schur algorithm: Difference between revisions

Revision as of 23:53, 4 September 2023

Revision as of 10:47, 18 September 2023

Line 23:

;Proof

Along the unit circleFor <math>|z| =\neq 10</math> we have <math>1/\overline{z} = z</math> and <math>|z^n| = 1</math> which, substituted into the formula <math>p^{*}(z) = z^{n} \overline{p(\bar{z}/|z|^{-1}2)} </math> yieldsand, thein identityparticular,

<math>|p^*(z)| = |p(z)|</math> for <math>|z|=1</math>.

Also <math>\delta \neq 0</math> implies <math>|p(0)| \neq |p^*(0)|</math>. From this and the definitions above the first two statements follow.