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On the convergence of some algorithms of binary or ternary machine arithmetic for calculations in imaginary quadratic fields
Bogdanov P.S.

 

Image Processing Systems Institute, Russian Academy of Sciences,

Samara State Aerospace University

 

DOI: 10.18287/0134-2452-2015-39-2-249-254

Full text of article: Russian language.

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Abstract:
The paper proves a number of statements that significantly reduce the complexity of proofs of the classification theorems for quasicanonical number systems in imaginary quadratic fields. Theorems on convergence of algorithms that implement the addition of algebraic integers in quasicanonical number systems are proved.

Keywords:
canonical numerical system, quasicanonical numerical system, norm division with remainder, equivalent numerical systems.

Citation:
Bogdanov PS. On the convergence of some algorithms of binary or ternary machine arithmetic for calculations in imaginary quadratic fields. Computer Optics 2015; 39(2): 249-254. DOI: 10.18287/0134-2452-2015-39-2-249-254.

References:

  1. Katai, I. Kanonische Zahlensysteme in der Theorie der Quadratischen Zahlen / I. Katai, B. Kovacs // Acta Scientiarum Mathematicarum (Szeged). – 1980. – Vol. 42. – P. 99-107.
  2. Katai, I. Canonical number systems in imaginary quadratic fields / I. Katai, B. Kovacs // Acta Mathematica Hungarica. – 1981. – Vol. 37. – P. 159-164.
  3. Kovacs, B. Canonical number systems in algebraic number fields / B. Kovacs // Acta Mathematica Hungarica. – 1981. – Vol. 37. – P. 405-407.
  4. Kovacs, A. Generalized binary number system / A. Kovacs // Annales Universitatis Scientiarum Budapest, Sectio Computatorica. – 2001. – Vol. 20. – P. 195-206.
  5. Bogdanov, P.S. Gaussian integers representation in pitti's number system // Computer Optics. – 2010. – Vol. 34(4). – P. 561-566. – ISSN 0134-2452. – (In Russian).
  6. Bogdanov, P.S. Classification of binary quasicanonical number systems in imaginary quadratic fields / P.S. Bogdanov, V.M. Chernov // Computer Optics. – 2013. – Vol. 37(3). – P. 391-400. – ISSN 0134-2452.
  7. Bogdanov, P.S. Classification of ternary quasicanonical number systems in imaginary quadratic fields and their application / P.S. Bogdanov, V.M. Chernov // Computer Optics. – 2014. – Vol. 38(1). – P. 139-147. – ISSN 0134-2452.
  8. Bogdanov, P.S. Dimension of Some fractal sets on hexagonal lattices / P.S. Bogdanov, V.M. Chernov // Computer Optics. – 2014. – Vol. 38(2). – P. 330-334. – ISSN 0134-2452.
  9. Borevich, Z.I. Number theory / Z.I. Borevich, I.R. Shafarevich. – Academic Press, 1986. – 434 p.
  10. Chernov, V.M. Arithmetical methods of synthesis of fast algorithms of Discrete orthogonal Transforms / V.M. Chernov. – Moscow: “Fizmatlit” Publisher, 2007. – 264 p. – (In Russian).
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