The Community for Technology Leaders
RSS Icon
Issue No.08 - Aug. (2012 vol.61)
pp: 1097-1109
Petr Kůrka , Charles University in Prague, Prague
We analyze the time complexity of exact real arithmetical algorithms in Möbius number systems. Using the methods of Ergodic theory, we associate to any Möbius number system its transaction quotient {\bf T}\ge 1 and show that the norm of the state matrix after n transactions is of the order {\bf T}^n. We argue that the Bimodular Möbius number system introduced in Kůrka has transaction quotient less than 1.2, so that it computes the arithmetical operations faster than any standard positional system.
Expansion subshifts, exact real arithmetical algorithms, emissions, absorptions, transactions.
Petr Kůrka, "Fast Arithmetical Algorithms in Möbius Number Systems", IEEE Transactions on Computers, vol.61, no. 8, pp. 1097-1109, Aug. 2012, doi:10.1109/TC.2012.87
[1] M.F. Barnsley, Fractals Everywhere. Morgan Kaufmann, 1993.
[2] R.W. Gosper, "Continued Fractions Arithmetic," Unpublished Manuscript,, 1977.
[3] R. Heckmann, "Big Integers and Complexity Issues in Exact Real Arithmetic," Electronic Notes Theoretical Computer Science, vol. 13, p. 69, 1998.
[4] A. Kazda, "Convergence in Möbius Number Systems," Integers, vol. 2, pp. 261-279, 2009.
[5] D.E. Knuth, The Art of Computer Programming, Seminumerical Algorithms, vol. 2, Addison-Wesley, 1981.
[6] P. Kornerup and D.W. Matula, "An Algorithm for Redundant Binary Bit-Pipelined Rational Arithmetic," IEEE Trans. Computers, vol. 39, no. 8, pp. 1106-1115, Aug. 1990.
[7] P. Kůrka, Topological and Symbolic Dynamics, vol. 11, Cours spécialisés, Société Mathématique de France, 2003.
[8] P. Kůrka, "A Symbolic Representation of the Real Möbius Group," Nonlinearity, vol. 21, pp. 613-623, 2008.
[9] P. Kůrka, "Möbius Number Systems with Sofic Subshifts," Nonlinearity, vol. 22, pp. 437-456, 2009.
[10] P. Kůrka, "Expansion of Rational Numbers in Möbius Number Systems," Dynamical Numbers: Interplay between Dynamical Systems and Number Theory, S. Kolyada, Y. Manin, and M. Moller, eds., vol. 532, pp. 67-82, Am. Math. Soc., 2010.
[11] P. Kůrka, "Stern-Brocot Graph in Möbius Number Systems," Nonlinearity, vol. 25, pp. 57-72, 2012.
[12] P. Kůrka and A. Kazda, "Möbius Number Systems Based on Interval Covers," Nonlinearity, vol. 23, pp. 1031-1046, 2010.
[13] D. Lind and B. Marcus, An Introduction to Symbolic Dynamics and Coding. Cambridge Univ. Press, 1995.
[14] R. Ma$\tilde{{\rm n}}$é, Ergodic Theory and Differentiable Dynamics, Ergebnisse der Mathematik und ihrer Grenzgebiete, Springer, 1983.
[15] P.J. Potts, "Exact Real Arithmetic Using Möbius Transformations," PhD thesis, Univ. of London, Imperial College, London, 1998.
[16] P.J. Potts, A. Edalat, and M.H. Escardó, "Semantics of Exact Real Computation," Proc. IEEE 12th Ann. Symp. in CS, pp. 248-257, 1997.
[17] J.E. Vuillemin, "Exact Real Computer Arithmetic with Continued Fractions," IEEE Trans. Computers, vol. 39, no. 8, pp. 1087-1105, Aug. 1990.
[18] K. Weihrauch, "EATCS Monographs on Theoretical Computer Science," Computable Analysis: An Introduction, Springer, 2000.
31 ms
(Ver 2.0)

Marketing Automation Platform Marketing Automation Tool