Fernando and Parlett observed that
the dqds algorithm for singular values
can be made extremely efficient with Rutishauser's choice of shift;
in particular it enjoys ``local'' (or one-step) cubic convergence
at the final stage of iteration, where a certain condition is to be
satisfied.
Their analysis is, however, rather heuristic and
what has been shown is not sufficient to ensure
asymptotic cubic convergence in the strict sense of the word.
The objective of this paper is to specify a concrete procedure
for the shift strategy and to prove with mathematical rigor that
the algorithm with this shift strategy always reaches
the ``final stage'' and enjoys asymptotic cubic convergence.