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A continued fraction algorithm for real algebraic numbers

Authors: David G. Cantor, Paul H. Galyean and Horst G. Zimmer
Journal: Math. Comp. 26 (1972), 785-791
MSC: Primary 12D10; Secondary 10F20
MathSciNet review: 0330118
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Abstract: Let a denote a real algebraic number that is a root of a polynomial $ f(x) \in {\text{Z}}[x]$. The purpose of this paper is to state an algorithm for finding the simple continued fraction expansion of $ \alpha $. Furthermore, an application of the algorithm to sign determination in real algebraic number fields is given.

References [Enhancements On Off] (What's this?)

  • [1] H. Kempfert, ``On sign determinations in real algebraic number fields,'' Numer. Math., v. 11, 1968, pp. 170-174. MR 37 #1355. MR 0225762 (37:1355)
  • [2] J. Lagrange, ``Sur la résolution des équations numériques,'' Oeuvres. Vol. 2, pp. 560-578.
  • [3] D. L. Smith, The Calculation of Simple Continued Fraction Expansions of Real Algebraic Numbers, Master Thesis, Ohio State University, Columbus, Ohio, 1969.
  • [4] J. V. Uspensky, Theory of Equations, McGraw-Hill, New York-Toronto-London, 1948.
  • [5] H. Zassenhaus, On the Continued Fraction Development of Real Irrational Algebraic Numbers, Ohio State University, Columbus, Ohio, 1968. (Unpublished.)
  • [6] H. Zassenhaus, ``A real root calculus,'' Computational Problems in Abstract Algebra, edited by J. Leech, Pergamon, Oxford, 1970. MR 0276205 (43:1953)

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Keywords: Continued fraction expansion, algorithm, discrimination of roots, irrational real algebraic numbers, PV numbers, binary search procedure, sign determination, mean value theorem
Article copyright: © Copyright 1972 American Mathematical Society

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