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as the degree of the polynomial increases.
If the coefficients are real and the polynomial has odd degree, then it must have at least one real root. To find this, use a real value of ''p''<sub>0</sub> as the initial guess and make ''q''<sub>0</sub> and ''r''<sub>0</sub>, etc., [[complex conjugate]] pairs. Then the iteration will preserve these properties; that is, ''p''<sub>''n''</sub> will always be real, and ''q''<sub>''n''</sub> and ''r''<sub>''n''</sub>, etc., will always be conjugate. In this way, the ''p''<sub>''n''</sub> will converge to a real root ''P''. Alternatively, make all of the initial guesses real; they will remain so.
== Example ==
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* Bo Jacoby, ''Nulpunkter for polynomier'', CAE-nyt (a periodical for Dansk CAE Gruppe [Danish CAE Group]), 1988.
* Agnethe Knudsen, ''Numeriske Metoder'' (lecture notes), Københavns Teknikum.
* Bo Jacoby, ''Numerisk løsning af ligninger'', Bygningsstatiske meddelelser (Published by Danish Society for Structural Science and Engineering) volume 63 no.
* {{cite book|last=Gourdon|first=Xavier|title=Combinatoire, Algorithmique et Geometrie des Polynomes|publisher=École Polytechnique|___location=Paris|year=1996|url=http://algo.inria.fr/gourdon/thesis.html|access-date=2006-08-22|archive-url=https://web.archive.org/web/20061028060132/http://algo.inria.fr/gourdon/thesis.html|archive-date=2006-10-28|url-status=dead}}
* [[Victor Pan]] (May 2002): [https://web.archive.org/web/20060907205721/http://www.cs.gc.cuny.edu/tr/techreport.php?id=26 ''Univariate Polynomial Root-Finding with Lower Computational Precision and Higher Convergence Rates'']. Tech-Report, City University of New York
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