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Issue No.11 - November (2010 vol.59)
pp: 1441-1454
Tomáš Brabec , Czech Technical University in Prague, Prague
Continued logarithms, as originally introduced by Gosper, represent a means for exact rational arithmetic, but their application to exact real arithmetic is limited by the uniqueness of their representation. This is quite unfortunate, as this representation seems promising for efficient hardware implementation. We propose an idea of making the representation redundant using speculative recognition of noncomputable cases. This approach solves the problem of real number computability, preserves most of the beneficial properties of continued logarithms, and only moderately affects complexity of arithmetic algorithms, thus, keeping the prospect of efficient implementation.
Computer arithmetic, representation of numbers, continued fraction, redundancy, computable real numbers, exact arithmetic.
Tomáš Brabec, "Speculatively Redundant Continued Logarithm Representation", IEEE Transactions on Computers, vol.59, no. 11, pp. 1441-1454, November 2010, doi:10.1109/TC.2010.110
[1] A. Khinchin, Continued Fractions, third ed. Univ. of Chicago Press, 1964, translated from Russian by P. Wynn, P. Noordhoff Ltd., 1963, and by H. Eagle, Univ. of Chicago Press, 1964.
[2] R.W. Gosper, "Continued Fraction Arithmetic," unpublished manuscript,, 1978.
[3] P. Gowland and D. Lester, "A Survey of Exact Arithmetic Implementations," Proc. Computability and Complexity in Analysis (CCA '00), J. Blanck, V. Brattka, and P. Hertling, eds., pp. 30-47, 2000.
[4] D.R. Lester, "Effective Continued Fractions," Proc. 15th IEEE Symp. Computer Arithmetic, pp. 163-170, 2001.
[5] H.J. Boehm, R. Cartwright, M. Riggle, and M.J. O'Donnell, "Exact Real Arithmetic: A Case Study in Higher Order Programming," Proc. ACM Symp. LISP and Functional Programming, pp. 162-173, 1986.
[6] O. Mencer, "Rational Arithmetic Units in Computer Systems," PhD thesis, EECS, Stanford Univ., , 2000.
[7] T. Brabec, "Hardware Implementation of Continued Logarithm Arithmetic," Proc. 12th GAMM IMACS Int'l Symp. Scientific Computing, Computer Arithmetic and Validated Numerics (SCAN '06) Conf. Post-Proc., 2006.
[8] P. Kornerup and D.W. Matula, "Finite Precision Lexicographic Continued Fraction Number Systems," Proc. Seventh IEEE Symp. Computer Arithmetic, pp. 207-214, 1985.
[9] M. Niqui, "Exact Arithmetic on the Stern-Brocot Tree," Technical Report NIII-R0325, Nijmeegs Instituut voor Informatica en Informateikunde, 2003.
[10] C. Mazenc, "On the Redundancy of Real Number Representation Systems," Research Report RR93-16, Laboratoire de l'Informatique du Parallélisme,, 1993.
[11] 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.
[12] T. Brabec, "Proof of Interior Containment," A Supplement to an Electronic Version of this Manuscript, http://doi. TC.2010.110, 2009.
[13] T. Brabec, "On Progress of Investigations in Continued Logarithm Arithmetic," Proc. Počítačové architektury a diagnostika 2007, pp. 61-66, pubpad2007extended.pdf, 2007.
[14] G.N. Raney, "On Continued Fractions and Finite Automata," Mathematische Annalen, vol. 206, no. 4, pp. 265-283, 1973.
[15] P.J. Potts, "Exact Real Arithmetic Using Möbius Transformations," PhD thesis, Imperial College, Univ. of London, http://www. , July 1998.
[16] R. Heckmann, "Contractivity of Linear Fractional Transformations," Theoretical Computer Science, vol. 279, nos. 1/2, pp. 65-82, 2002.
[17] R. Heckmann, "How Many Argument Digits are Needed to Produce n Result Digits?" Electronic Notes in Theoretical Computer Science, vol. 24, pp. 13-33, 1999.
[18] K.-I. Ko, "On the Continued Fraction Representation of Computable Real Numbers," Theoretical Computer Science, vol. 47, no. 3, pp. 299-313, 1986.
[19] T. Brabec, "Redundant Cont. Log. Representation: Proof of Contractivity," unpublished paper, pubclproof08.pdf, 2008.
[20] T. Brabec, "Continued Logarithms," manuscript in preparation, pubclreport08.pdf, 2008.
[21] R.W. Gosper, "Item 101 in Hakmem," AIM239, MIT, pp. 37-44, 2AIM-239.pdf, Apr. 1972.
[22] T. Brabec, "Quantitative Results," A Supplement to an Electronic Version of this Manuscript, TC.2010.110, 2009.
[23] Y.-K. Kwok and I. Ahmad, "Static Scheduling Algorithms for Allocating Directed Task Graphs to Multiprocessors," ACM Computing Surveys, vol. 31, no. 4, pp. 406-471, 1999.
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