Lehmer–Schur algorithm: Difference between revisions

Content deleted Content added
Line 2:
 
==Schur-Cohn algorithm==
This [[algorithm]] allows one to find the distribution of the roots of a complex polynomial with respect to the [[unit circle]] in the complex plane.<ref>{{cite journal |last1=Cohn |first1=A |title=Uber die Anzahl der Wurzeln einer algebraischen Gleichung in einem Kreise. |journal=Math. Z. |date=1922 |volume=14 |pages=110–148 |doi=10.1007/BF01215894 |url=http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN266833020_0014&DMDID=dmdlog10|hdl=10338.dmlcz/102550 |hdl-access=free }}</ref><ref name="Henrici">{{cite book |last1=Henrici |first1=Peter |title=Applied and computational complex analysis. Volume I: Power series- integration-conformal mapping-___location of zeros. |date=1988 |publisher=New York etc.: John Wiley |isbn=0-471-60841-6 |pages=xv + 682 |edition= Repr. of the orig., publ. 1974 by John Wiley \& Sons Ltd., Paperback}}</ref><ref>{{cite book |last1=Marden |first1=Morris |title=The geometry of the zeros of a polynomial in a complex variable. |date=1949 |publisher=Mathematical Surveys. No. 3. New York: American Mathematical Society (AMS). |page=148 }}</ref> It is based on two auxiliary polynomials, introduced by Schur.<ref>{{cite journal |last1=Schur |first1=I |title=Über Potenzreihen, die im Innern des Einheitskreises beschränkt sind. |journal=Journal für die reine und angewandte Mathematik |date=1917 |volume=1917 |issue=147 |pages=205–232 |doi=10.1515/crll.1917.147.205 |url=http://gdz.sub.uni-goettingen.de/dms/load/img/?PID=GDZPPN00216860X}}</ref>
 
For a complex polynomial <math>p</math> of [[degree of a polynomial|degree]] <math>n</math> its ''reciprocal adjoint polynomial'' <math>p^{*}</math> is defined by <math>p^{*}(z) = z^{n}\overline{p(\bar{z}^{-1})} </math> and its ''SchurtransformSchur Transform'', <math>Tp</math>, by
 
:<math>