A parallel block cyclic reduction algorithm for the fast solution of elliptic equations

  • E. Gallopoulos
  • Y. Saad
Session 6B: Parallel Numeric Methods
Part of the Lecture Notes in Computer Science book series (LNCS, volume 297)


This paper presents an adaptation of the Block Cyclic Reduction (BCR) algorithm for a multi-vector processor. The main bottleneck of BCR lies in the solution of linear systems whose coefficient matrix is the product of tridiagonal matrices. This bottleneck is handled by expressing the rational function corresponding to the inverse of this product as a sum of elementary fractions. As a result the solution of this system leads to parallel solutions of tridiagonal systems. Numerical experiments performed on an Alliant FX/8 are reported.


Fast Fourier Transform Outer Loop Matrix Polynomial Vector Length Elliptic Partial Differential Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [BuGN70]
    B. Buzbee, G. Golub, and C. Nielson, "On direct methods for solving Poisson's equation" SIAM J. Numer. Anal, vol. 7, pp. 627–656, (December 1970).Google Scholar
  2. [Hock70]
    R. Hockney, "The Potential Calculation and Some Applications", in Methods Comput. Phys., v. 9. pp. 135–211, Academic Press, (1970).Google Scholar
  3. [OrVo85]
    J. Ortega and R. Voigt, "Partial differential equations on vector and parallel computers", SIAM Review, pp. 213–240, (June 1985).Google Scholar
  4. [RiHD81]
    J. Rice, E. Houstis and W. Dyksen, "A population of linear, second order elliptic partial differential equations on rectangular domains. Part 1", Math. Comp., v. 36, pp. 479–484, (1981).Google Scholar
  5. [Saad87]
    Y. Saad, "On the design of parallel numerical methods in message passing and shared-memory environments", Proc. International Seminar on Scientific Supercomputer, Paris, (2–6 February 1987).Google Scholar
  6. [SaCK76]
    A. H. Sameh, S. C. Chen, D. J. Kuck, "Parallel Poisson and biharmonic solvers", Computing, vol. 17, pp. 219–230 (1976).Google Scholar
  7. [Swar77]
    P. N. Swarztrauber, "The methods of cyclic reduction, Fourier analysis and the FACR algorithm for the discrete solution of Poisson's equation on a rectangle", SIAM Review, v. 19, pp. 490–501, (July 1977).Google Scholar
  8. [Swar84]
    P. N. Swarztrauber, "Fast Poisson solvers", in Studies in Numerical Analysis, G. H. Golub ed., pp. 319–369, Mathematical Association of America, (1984).Google Scholar
  9. [Swar87]
    P. N. Swarztrauber, "Approximate cyclic reduction for solving Poisson's equation", SIAM J. Sci. Stat. Comput., v. 8, pp. 199–209, (May 1987).Google Scholar
  10. [SwSw75]
    P. Swarztrauber and R. Sweet, "Efficient Fortran subprograms for the solution of elliptic partial differential equations", NCAR Technical Note IA-109, Boulder, (July 1975).Google Scholar
  11. [Swee77]
    R. A. Sweet, "A cyclic reduction algorithm for solving block tridiagonal systems of arbitrary dimension", SIAM J. Numer. Anal., vol. 14, pp. 707–720, (September 1977).Google Scholar
  12. [Temp80]
    C. Temperton, "On the FACR(l) algorithm for the discrete Poisson equation", J. of Comp. Physics, v. 34, pp. 314–329, (1980).Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1988

Authors and Affiliations

  • E. Gallopoulos
    • 1
  • Y. Saad
    • 1
  1. 1.Center for Supercomputing Research and DevelopmentUniversity of Illinois at Urbana ChampaignUrbanaU.S.A.

Personalised recommendations