Parallel Computation of the SVD of a Matrix Product

  • Josée M. Claver
  • Manuel Mollar
  • Vicente Hernández
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1697)


In this paper we studya parallel algorithm for computing the singular value decomposition (SVD) of a product of two matrices on message passing multiprocessors. This algorithm is related to the classical Golub-Kahan method for computing the SVD of a single matrix and the recent work carried out byGolu b et al. for computing the SVD of a general matrix product/quotient. The experimental results of our parallel algorithm, obtained on a network of PCs and a SUN Enterprise 4000, show high performances and scalabilityfor large order matrices.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Anderson, E., Bai, Z., Bischof, C., Demmel, J., Dongarra, J., Du Croz, J., Greenbaum, A., Hammarling, S., Mckenney, A., Ostrouchov, S., Sorensen, D.: LAPACK User’s Guide, Release 1.0., SIAM, Philadelphia (1992).Google Scholar
  2. 2.
    Bai, Z: A parallel algorithm for computing the generalized singular value decomposition, Journal of Parallel and Distributed Computing 20 (1994) 280–288.zbMATHCrossRefGoogle Scholar
  3. 3.
    Brent, R., Luk, F. and van Loan, C.: Computation of the generalized singular value decomposition using mesh connected processors, Proc. SPIE Vol. 431, Real time signal processing VI (1983) 66–71.Google Scholar
  4. 4.
    Blackford, L., Choi, J., D’Azevedo, E., Demmel, J., Dhillon, I., Dongarra, L., Hammarling, S., Henry, G., Petitet, A., Stanley, K., Walker, D., Whaley, R.: SCALAPACK User’s Guide, SIAM (1997).Google Scholar
  5. 5.
    De Moor, B. and Golub, G. H.: Generalized singular value decompositions: A proposal for a standardized nomenclature, Num. Anal. Proj. Report 89-04, Comput. Sci. Dept., Stanford University(1989).Google Scholar
  6. 6.
    De Moor, B.: On the structure and geometryof the PSVD, Num. Anal. Project, NA-89-05, Comput. Sci. Dept., Stanford University (1989).Google Scholar
  7. 7.
    Demmel, J., Kahan, W.: Accurate singular values of bidiagonal matrices, SIAM J. Sci. Stat. Comput. 11 (1990) 873–912.zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Fernando, K. and Hammarling, S.: A generalized singular value decomposition for a product of two matrices and balanced realization, NAG Technical Report TR1/87, Oxford (1987).Google Scholar
  9. 9.
    Fernando, K. and Parlett, B.: Accurate singular values and differential qd algorithms. Numerische Mathematik 67 (1994) 191–229.zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Golub, G., W. Kahan, Calculation of the singular values and the pseudoinverse of a matrix, SIAM J. Numer. Anal. 2 (1965) 205–224.CrossRefMathSciNetGoogle Scholar
  11. 11.
    Golub, G., Reinsch, W.: Singular value decomposition and the least square solution, Numer. Mathematik 14, (1970) 403–420.zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Golub, G., Sølna, K, and van Dooren, P.: Computing the SVD of a General Matrix Product/Quotient, submitted to SIAM J. on Matrix Anal. & Appl.,(1997).Google Scholar
  13. 13.
    Golub, G., Van Loan, C.: Matrix Computations, North Oxford Academic, Oxford (1983).zbMATHGoogle Scholar
  14. 14.
    Heat, M., Laub, A., Paige, C., Ward, R.: Computing the singular value decomposition of a product of two matrices, SIAM J. Sci. Stat. Comput. 7 (1986) 1147–1159.CrossRefGoogle Scholar
  15. 15.
    Laub, A., Heat, M., Paige, G., Ward, R.: Computation of system balancing transformations and other applications of simultaneous diagonalization algorithms, IEEE Trans. AC 32 (1987) 115–122.zbMATHGoogle Scholar
  16. 16.
    Mollar, M., Hernández, V.: Computing the singular values of the product of two matrices in distributed memory multiprocessors, Proc. 4th Euromicro Workshop on Parallel and Distributed Computation, Braga (1996) 15–21.Google Scholar
  17. 17.
    Mollar, M., Hernández, V.: A parallel implementation of the singular value decomposition of the product of triangular matrices, 1st NICONET Workshop, Valencia (1998)Google Scholar
  18. 18.
    Moore, B.: Principal component analysis in linear systems: Controlability, observability, and model reduction, IEEE Trans. AC 26 (1981) 100–105.Google Scholar
  19. 19.
    Paige, C., Sanders, M.: Towards a generalized singular value decomposition, SIAM J. Numer. Anal. 18 (1981) 398–405.zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Van Loan, C.: Generalizing the singular value decomposition, SIAM J. Numer. Anal. 13 (1976) 76–83.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Josée M. Claver
    • 1
  • Manuel Mollar
    • 1
  • Vicente Hernández
    • 2
  1. 1.Dpto. de InformáticaUniv. Jaume ICastellónSpain
  2. 2.Dpto. de Sistemas Informáticos y ComputaciónUniversidad Politécnica de ValenciaValenciaSpain

Personalised recommendations