Numerische Mathematik

, Volume 67, Issue 2, pp 191–229

Accurate singular values and differential qd algorithms

  • K. Vince Fernando
  • Beresford N. Parlett

DOI: 10.1007/s002110050024

Cite this article as:
Fernando, K. & Parlett, B. Numer. Math. (1994) 67: 191. doi:10.1007/s002110050024


We have discovered a new implementation of the qd algorithm that has a far wider domain of stability than Rutishauser's version. Our algorithm was developed from an examination of the {Cholesky~LR} transformation and can be adapted to parallel computation in stark contrast to traditional qd. Our algorithm also yields useful a posteriori upper and lower bounds on the smallest singular value of a bidiagonal matrix. The zero-shift bidiagonal QR of Demmel and Kahan computes the smallest singular values to maximal relative accuracy and the others to maximal absolute accuracy with little or no degradation in efficiency when compared with the LINPACK code. Our algorithm obtains maximal relative accuracy for all the singular values and runs at least four times faster than the LINPACK code.

Mathematics Subject Classification (1991): 65F15

Copyright information

© Springer-Verlag Berlin Heidelberg 1994

Authors and Affiliations

  • K. Vince Fernando
    • 1
  • Beresford N. Parlett
    • 2
  1. 1.NAG Ltd, Wilkinson House, Jordan Hill, Oxford OX2 8DR, UK GB
  2. 2.Department of Mathematics, University of California, Berkeley, CA 94720, USA US