I could only observe that *A*^{T}*A* and *A*^{T}*WA* have more or less the same
condition number during the IRLS algorithm; consequently, the weighted
least-squares problems are also ill-conditioned if the original power spectrum
of the filter *a* is band-limited. However, I have no general
results concerning the repartition of the eigenvalues of *A*^{T}*WA*.

The choice of is not completely free. The synthetic examples I
used showed that the more ill-conditioned the initial matrix *A*^{T}*A* is, the
larger should be taken, otherwise the IRLS algorithm does not
converge. For example, with a condition number equal to 1000,
I took , and the convergence (rate
1/10000) was reached in 10 iterations; with a condition number equal to
10000, I took , even with double
precision. Increasing makes *W* closer to the identity matrix,
in which case the problem corresponds more to a *L ^{2}* minimization: we would be
solving a mixed

In conclusion, I may expect the convergence of the CG algorithms to be
slow if the power spectrum of the filter *a* contains some small values.
Despite the suggestion to limit the number of iterations in order
to cope with the ill-conditioning, I prefer to increase their number, to be
closer to the numerical limits. However, the speed of convergence is greatly
increased if it is possible to give a first estimate of the unknown *x*:
that is the case in predictive deconvolution, where the first estimate would
be the *L ^{2}*-Wiener filter.

1/13/1998