Advertisement

Incomplete Choleski Conjugate Gradient on the Cyber 203/205

  • Eugene L. Poole
  • James M. Ortega

Abstract

We consider in this paper the Incomplete Cholesky Conjugate Gradient (ICCG) method on the CDC Cyber 203/205 vector computers for the solution of an NxN system of linear equations Ax=b. We assume that the matrix A is large, sparse, and symmetric positive definite with non-zero elements lying along a few diagonals of the matrix, such as arises in the solution of elliptic partial differential equations by finite difference or finite element discretizations. Results are given for two model problems, run on a Cyber 203 at NASA — Langley Research Center. In the sequel, we shall refer to the Cyber 203 and 205 as the Cyber 200 unless there is a reason to differentiate between them.

Keywords

Model Problem Conjugate Gradient Method Vector Length Precondition Conjugate Gradient Matrix Vector Multiplication 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Adams, L. [ 1983a ], “Iterative Algorithms for Large Sparse Linear Systems on Parallel Computers, ” Ph.D. Dissertation, Applied Mathematics, University of Virginia.Google Scholar
  2. Adams, L. [1983b], “An M-Step Preconditioned Conjugate Gradient Method for Parallel Computation,” Proc. 1983 Intern. Conf. Par. Proc., pp. 36– 43.Google Scholar
  3. Adams, L. and Ortega, J. [1982], “A Multi-Color SOR Method for Parallel Computation, ” Proc. 1982 Intern. Conf. Par. Proc., pp. 53– 56.Google Scholar
  4. Horowitz, E. and Salmi, S. [ 1978 ], Fundamentals of Computer Algorithms, Computer Science Press, Rockville, Maryland.Google Scholar
  5. Kershaw, D. [1982], “Solution of Single Tridiagonal Linear Systems and Vectorization of the ICCG Algorithm on the CRAY-1, ” in Parallel Computations, G. Rodrigue, Editor, Academic Press, New York, N.Y. pp. 85–99.Google Scholar
  6. Madsen, N., Rodrigue, G. and Karush, J. [1976], “Matrix Multiplication by Diagonals on Vector/Parallel Processors, ” Infor. Proc. Letters 5, pp. 41–45.Google Scholar
  7. Lambiotte, J. [ 1974 ], “The Solution of Linear Equations On A Vector Computer, ” Ph.D. Dissertation, Applied Mathematics, University of Virginia.Google Scholar
  8. Lichnewsky, A. [1984], “Some Vector and Parallel Implementations for Preconditioned Conjugate Gradient Algorithms. ” Proceedings of the NATO Workshop on High-Speed Computations. NATO ASI Series, v. F7, pp. 343–359.Google Scholar
  9. Meijerink, J. and Van der Vorst, H. [1977], “An Iterative Solution for Linear Systems of Which the Coefficient Matrix is a Symmetric M-Matrix, ” Math. Comp. 31, pp. 148–162.Google Scholar
  10. Meijerink, J. and Van der Vorst, H. [1981], “Guidelines for the Usage of Incomplete Decompositions in Solving Sets of Linear Equations, ” J. Comp. Phys. 44, pp. 134–155. Google Scholar
  11. Schreiber, R. and Tang, W. [ 1982 ], “Vectorizing the Conjugate Gradient Method,” Proceedings Symposium Cyber 205 Applications. Ft. Collins, CO.Google Scholar

Copyright information

© Plenum Press, New York 1985

Authors and Affiliations

  • Eugene L. Poole
    • 1
  • James M. Ortega
    • 1
  1. 1.University of VirginiaUSA

Personalised recommendations