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The Jacobi-Perron algorithm in integer form


Authors: M. D. Hendy and N. S. Jeans
Journal: Math. Comp. 36 (1981), 565-574
MSC: Primary 10A30; Secondary 12A45
DOI: https://doi.org/10.1090/S0025-5718-1981-0606514-X
MathSciNet review: 606514
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Abstract | References | Similar Articles | Additional Information

Abstract: We present an alternative expression of the Jacobi-Perron algorithm on a set of $ n - 1$ independent numbers of an algebraic number field of degree n, where computation of real valued (nonrational) numbers is avoided. In some instances this saves the need to compute with high levels of precision. We also demonstrate a necessary and sufficient condition for the algorithm to cycle. The paper is accompanied by several numerical examples.


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

  • [1] William W. Adams, "On a relationship between the convergents of the nearest integer and regular continued fractions," Math. Comp., v. 33, 1979, pp. 1321-1331. MR 537978 (82g:10078)
  • [2] Leon Bernstein, The Jacobi-Perron Algorithm: Its Theory and Applications, Lecture Notes in Math., Vol. 207, Springer-Verlag, New York, 1971. MR 0285478 (44:2696)
  • [3] Leon Bernstein, "A 3-dimensional periodic Jacobi-Perron Algorithm of length 8," J. Number Theory, v. 4, 1972, pp. 48-61. MR 0294259 (45:3328)
  • [4] G. Chrystal, Algebra, Part II, 2nd ed., A. and C. Black Ltd., London, 1931.
  • [5] M. D. Hendy, "Applications of a continued fraction algorithm to some class number problems," Math, Comp., v. 28, 1974, pp. 267-277. MR 0330102 (48:8440)
  • [6] N. S. Jeans & M. D. Hendy, "Determining the fundamental unit of a pure cubic field given any unit," Math Comp., v. 32, 1978, pp. 925-935. MR 0472761 (57:12451)
  • [7] O. Perron, "Der Jacobische Kettenalgorithmns in einem Kubischen Zahlenkörper," Bayer. Akad. Wiss. Math.-Natur. Kl. Abh., 1971, pp. 13-49. MR 0556654 (58:27725a)
  • [8] H. M. Stark, "An explanation of some exotic continued fractions found by Brillhart," Computers in Number Theory (A. O. L. Atkin & B. J. Birch, Eds.), Academic Press, London, 1971, pp. 21-35. MR 0337801 (49:2570)
  • [9] Hideo Wada, "A table of fundamental units of purely cubic fields," Proc. Japan Acad., v. 46, 1970, pp. 1135-1140. MR 0294292 (45:3361)
  • [10] H. Williams & P. Buhr, "Calculations of the regulator of $ Q(\surd d)$ by use of the nearest integer continued fraction algorithm," Math Comp., v. 33, 1979, pp. 369-381. MR 514833 (80e:12003)

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Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1981-0606514-X
Keywords: Jacobi-Perron algorithm, multiprecision arithmetic, continued fractions, fundamental unit, cubic fields
Article copyright: © Copyright 1981 American Mathematical Society

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