Abstract
The speed of integer and rational arithmetic increases significantly by systolic implementation on a SIMD architecture. For multiplication of integers one obtains linear speed-up (up to 29 times), using a serial-parallel scheme. A two-dimensional algorithm for multiplication of polynomials gives half-linear speed-up (up to 383 times). We also implement multiprecision rational arithmetic using known systolic algorithms for addition and multiplication, as well as recent algorithms for exact division and GCD computation. All algorithms work in “least-significant digits first” pipelined manner, hence they can be well aggregated together. The practical experiments show that the timings depend linearly on the input length, demonstrating the effectiveness of the systolic paradigm for multiple precision arithmetic.
Supported by the Austrian Fonds zur Förderung der wissenschaftlichen Forschung, project P10002 MAT
Preview
Unable to display preview. Download preview PDF.
References
R. Beardsworth. On the application of array processors to symbol manipulation. In SYMSAC'81, 1981.
A. D. Booth. A signed binary multiplication technique. Q. J. Mech. Appl. Math., 4:236–240, 1951.
R. P. Brent and H. T. Kung. A systolic algorithm for integer GCD computation. In K. Hwang, editor, Procs. of the 7th Symp. on Computer Arithmetic, pages 118–125. IEEE Computer Society, June 1985.
B. Buchberger. Gröbner Bases: An Algorithmic Method in Polynomial Ideal Theory. In Bose and Reidel, editors, Recent trends in Multidimensional Systems, pages 184–232, Dordrecht-Boston-Lancaster, 1985. D. Reidel Publishing Company.
P. Henrici. A subroutine for computations with rational numbers. Journal of the ACM, 3:6–9, 1956.
T. Jebelean. A generalization of the binary GCD algorithm. In M. Bronstein, editor, ISSAC93: International Symposium on Symbolic and Algebraic Computation, pages 111–116, Kiev, Ukraine, July 1993. ACM Press.
T. Jebelean. An algorithm for exact division. Journal of Symbolic Computation, 15(2):169–180, February 1993.
T. Jebelean. Systolic Algorithms for Exact Division. In Workshop on Fine Grain and Massive Parallelism, pages 40–50, Dresden, Germany, April 1993. Published in Mitteilngen-Gesellschaft für Informatik e. V. Parallel Algorithmen und Rechnerstrukturen, Nr. 12, July 1993.
T. Jebelean. Systolic algorithms for long integer GCD computation. In J. Volkert B. Buchberger, editor, CONPAR 94 — VAPP VI, Linz, Austria, September, pages 241–252. Springer Verlag LNCS 854, 1994.
D. H. Lehmer. Euclid's algorithm for large numbers. Am. Math. Mon., 45:227–233, 1938.
J. Stein. Computational problems associated with Racah algebra. J. Comp. Phys., 1:397–405, 1967.
E. E. Swartzlander, editor. Computer Arithmetic, volume 2. IEEE Computer Society Press, 1990.
K. S. Trivedi and M. D. Ercegovac. On-line algorithms for division and multiplication. IEEE Trans. on Computers, C-26(7):681–687, 1977.
K. Weber. The accelerated integer GCD algorithm. ACM Trans. on Math. Software, 21(1):111–122, March 1995.
D. Weeks. Adaptation of SAC-1 algorithms for an SIMD machine. In J. Della Dora and J. Fitch, editors, Computer Algebra and Parallelism, pages 167–177. Academic Press, 1989.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1996 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Jebelean, T. (1996). Integer and rational arithmetic on masPar. In: Calmet, J., Limongelli, C. (eds) Design and Implementation of Symbolic Computation Systems. DISCO 1996. Lecture Notes in Computer Science, vol 1128. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-61697-7_15
Download citation
DOI: https://doi.org/10.1007/3-540-61697-7_15
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-61697-9
Online ISBN: 978-3-540-70635-9
eBook Packages: Springer Book Archive