Accurate singular values and differential qd algorithms

Abstract

We have discovered a new implementation of theqd 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 yieldsuseful a posteriori upper and lower bounds on the smallest singularvalue of a bidiagonal matrix.The zero-shift bidiagonal QR of Demmel and Kahan computes the smallest singularvalues to maximal relative accuracy and the others to maximal absolute accuracywith little or no degradation in efficiency when compared with theLINPACK code. Our algorithm obtains maximal relative accuracy for allthe singular values and runs at least four times faster than the LINPACK code. © 1994, Springer-Verlag Berlin Heidelberg. All rights reserved.