Massively parallel symbolic computation

ABSTRACT

A massively parallel processor proves to be a powerful tool for manipulating the very large Poisson series encountered in non-linear dynamics. Exploiting the algebraic structure of Poisson series leads quite naturally to parallel data structures and algorithms for symbolic manipulation. Exercising the parallel symbolic processor on the solution of Kepler's equation reveals the need to reexamine the serial computational methods traditionally applied to problems in dynamics.

REFERENCES

• 1: Agnese, J.-C. Logiciel MSTN: Manipulation de S~ries Trigonomatriques. Teclmical Report CT/DTi/MS/MN/262 (1984), Centre NationM d't~tudes Spatiales {CNES).
• 2: Broucke, R., and Gaxthwaite, K. A Progranmling System for Analytical Series Expansions on a Computer. Celestial Mechanics 1 (1969), 271-284.
• 3: Cayley, A. Tables of the Developments of Functions in the Theory of Elliptic Motion. Memoirs of the Royal Astronomicai Society 29, 191-306. Reprinted in The Collected 1%qathematicaI Papers of Arthur Cayley, Vol. 3. Cambridge University Press, 1890, pp. 360-474.
• 4: Connection Machine Model CM-2 Technical Summary. Thinking Machines Corporation, Cambridge, Mass., 1987.
• 5: Danby A. J., Deprit, A., and Rom A. The Symbolic Maniptdation of Poisson Series. Boeing Docmnent D1-82-0481 (1965), Boeing Scientific Resem'ch Laboratory.
• 6: Dasenbrock, R. R. A FORTRAN-Based Program for Computerized Algebraic Manipulation. NRL Report 8611 (1982), U.S. Naval Research Laboratory.
• 7: Deprit, A. Canonical Trm~sformations Depending on a Small Parameter. Celestial Mechanics 1 (1969), 12-30.
• 8: Deprit, A., and Miller, Brace R. Normalization in the Face of hltegrability. Annals of the New York Academy of Sci* ences 536 (1988), 101-126.
• 9: Deprit, A., attd Miller, Bruce R. Simplify or Perish. To appeal' in Proceeding~ of the h~,ernational Astro~omical Uniol~ ColloquitJm ~109 in Celestial Mechanics.
• 10: W. Daniel Hillis, The connection machine, MIT Press, Cambridge, MA, 1986
• 11: Jefferys, W. H. A Fortrm~ Based Lisp Processor for Poisson Series. Celestial Mechanics 2 (1970), 474-480.
• 12: Kusmin, A. V. SASM: Operations on Series. The Algorithms of Celestial Mechanics (Institute of Theoretical Astronomy, U.S.S.R. Academy of Sciences, Leningrad) 40 (1982), 54.
• 13: MACSYMA Reference Manual: Version 13. Symbolic.s, Inc., Cambridge, Mass., 1988.
• 14: Meyer, K.R. Bifurcations and Stability by Lie Transformations. To appeax in Proceedings of the /nt~ernationa/Astronomical Union Colloquium ~109 in Celestial Mechanics.
• 15: Paris Reference Manual. Thinking Machines Corporation, Cambridge, Mass., 1988.
• 16: Rom, A. Mechanized Algebraic Operations (MAO). Celestial Mechan/cs 1 (1970), 301-319.
• 17: *Lisp Reference Manual. Version 5.0. Thinking Machines Corporation, Cambridge, Mass., 1988.
• 18: *Lisp Release Notes. Thinking Machines Corporation, Cambridge, Mass., 1988.
• 19: Guy L. Steele, Jr., Common LISP: the language, Digital Press, Newton, MA, 1984

INDEX TERMS

Primary Classification:
  F. Theory of Computation
  F.1 COMPUTATION BY ABSTRACT DEVICES
      F.1.2 Modes of Computation
          Subjects: Parallelism and concurrency

Additional Classification:
  C. Computer Systems Organization
  C.1 PROCESSOR ARCHITECTURES
      C.1.3 Other Architecture Styles
          Subjects: Stack-oriented processors**

  E. Data

  G. Mathematics of Computing
  G.1 NUMERICAL ANALYSIS

General Terms:
Algorithms, Design, Theory