Parallel algorithms for triangular sylvester equations: Design, scheduling and scalability issues

  • Peter Poromaa
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1541)


A new scalable algorithm for solving (quasi)triangular Sylvester equations on a logical 2D-toroidal processor grid is presented. Based on a performance model of the algorithm, a static scheduler chosing the optimal processor grid and block sizes for a rectangular block scatter (RBS) mapping of matrices is incorporated in the algorithm.

Scalability properties implying good scalability of the algorithm are also derived from the performance model. Performance results from our ScaLAPACK-like implementations on an 64 processor IBM SP verify the scalability potential of the algorithm.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    E. Anderson, Z. Bai, C. Bischof, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, A. Hammarling, A. McKenney, S. Ostrouchov, and D Sorensen. LAPACK Users Guide, Release 2.0. SIAM Press, September 1994.Google Scholar
  2. 2.
    Z. Bai and J. Demmel. On Swapping Diagonal Blocks in Real Schur Form. Linear Alg. Appl., 186:73–95, 1993.MATHMathSciNetCrossRefGoogle Scholar
  3. 2.
    Z. Bai, J. Demmel, and A. McKenney. On Computing Condition Numbers for the Nonsymmetric Eigenproblem. ACM Trans. Math. Softw., 19:202–223, October 1993.MATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    S. P. Bhattacharyya and E. de Souza. Pole Assignment via Sylvester’s Equation. Systems and Control Letters, 1:261–263, 1982.MATHCrossRefGoogle Scholar
  5. 5.
    J. Choi, J. J. Dongarra, R. Pozo, and D. W. Walker. ScaLAPACK: A Scalable Linear Algebra Library for Distributed Memory Concurrent Computers. Progress paper, Oak Ridge National Laboratory and University of Tennesee, 1991.Google Scholar
  6. 6.
    B. N. Datta. Parallel and Large-Scale Matrix Computations in Control: Some Ideas. Linear Alg. Appl., 121:243–264, 1989.MATHCrossRefGoogle Scholar
  7. 7.
    J. Dongarra and C. Whaley. A User’s Guide to the BLACS v1.0. LAPACK working note 94, Department of Computer Science, University of Tennessee, Knoxville, TN 37996, June 1995.Google Scholar
  8. 8.
    J. J. Dongarra and D. W. Walker. Software libraries for linear algebra computations on high performance computers. SIAM Review, 37(2):151–180, June 1995.MathSciNetCrossRefGoogle Scholar
  9. 9.
    N. J. Higham. Perturbation theory and backward error analysis for AX-XB=C. BIT, 33(1):124–136, 1993.MATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    B. Kågström. A Direct Method for Reordering Eigenvalues in the Generalized Real Schur Form of a Regular Matrix Pair (A, B). In G. H. Golub M. S. Moonen and B.L.R. De Moor, editors, Linear Algebra for Large Scale and Real-Time Applications, pages 195–218. Kluwer Academic Publicher, 1993.Google Scholar
  11. 11.
    B. Kågström. A Perturbation Analysis of the Generalized Sylvester Equation. SIAM J. MATRIX ANAL. APPL., 15(4):1045–1060, October 1994.MATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    B. Kågström and P. Poromaa. Distributed Block Algorithms for the Triangular Sylvester Equation with Condition Estimator. In F. Andre’ and J.P. Verjus, editors, Hypercube and Distributed Computers, pages 233–248. North-Holland, 1989.Google Scholar
  13. 13.
    B. Kågström and P. Poromaa. Computing Eigenspaces with Specified Eigenvalues of a Regular Matrix Pair (A, B) and Condition Estimation: Theory Algorithms and Software. Numerical Algorithms, 12:369–407, 1996.MATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    B. K⇘gström and P. Poromaa. LAPACK-Style Algorithms and Software for Solving the Generalized Sylvester Equation and Estimating the Separation between Regular Matrix Pairs. ACM Transactions on Mathematical Software, 22(1):78–103, 1996.CrossRefGoogle Scholar
  15. 15.
    V. Kumar, A. Grama, and G. Karpys. Introduction to Parallel Computing. Benjamin/Cummings, September 1994.Google Scholar
  16. 16.
    R. Mathias. Condition Estimation for Matrix Functions via the Schur Decomposition. Technical report. Department of Matematics, College of William and Mary, Williamsburg, USA, October 1994.Google Scholar
  17. 17.
    P. Poromaa. Design Implementation and Performance for the Triangular Sylvester Equation with Condition Estimators. UMINF-189-90, Licentiate Thesis, Institute of Information Processing, Department of Computing Science, University of Umeå, Sweden, November 1990.Google Scholar
  18. 18.
    P. Poromaa. On Efficient and Robust Estimators for the Separation between two Regular Matrix Pairs with Applications in Condition Estimation. UMINF-95-05, Department of Computing Science, Umeå Uneversity, Sweden, August 1995.Google Scholar
  19. 19.
    P. Poromaa. High Performance Computing: Algorithms and Library Software for Sylvester Equations and Certain Eigenvalue Problems with Applications in Condition Estimators, PhD Thesis. UMINF-97.16, Department of Computing Science, University of Umeå, Sweden, May 1997.Google Scholar
  20. 20.
    G.W. Stewart. Error and perturbation bounds for subspaces associated with certain eigenvalue problems. SIAM Review, 15:727–764, 1973.MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • Peter Poromaa
    • 1
  1. 1.Department of Computing ScienceUmeå UniversityUmeåSweden

Personalised recommendations