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 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.

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Oskar Perron,*Der Jacobi’sche Kettenalgorithmus in einem kubischen Zahlenkörper. II*, Bayer. Akad. Wiss. Math.-Natur. Kl. S.-B.**Abt. II**(1973), 9–22 (1974) (German). MR**0556655****[8]**H. M. Stark,*An explanation of some exotic continued fractions found by Brillhart*, Computers in number theory (Proc. Sci. Res. Council Atlas Sympos. No. 2, Oxford, 1969) Academic Press, London, 1971, pp. 21–35. MR**0337801****[9]**Hideo Wada,*A table of fundamental units of purely cubic fields*, Proc. Japan Acad.**46**(1970), 1135–1140. MR**0294292****[10]**H. C. Williams and P. A. Buhr,*Calculation of the regulator of 𝑄(√𝐷) by use of the nearest integer continued fraction algorithm*, Math. Comp.**33**(1979), no. 145, 369–381. MR**514833**, https://doi.org/10.1090/S0025-5718-1979-0514833-1

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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