The matrix *A*^{T}*WA* is symmetric and positive. However, it does not have
a particular structure (such as the Toeplitz structure of *A*^{T}*A* in
deconvolution). The computations of *A*^{T}*WA* and of its inverse are very
expensive, in time and in memory. That is why several authors
(Yarlagadda et al., 1985; Gersztenkorn et al., 1986; Scales et al., 1988)
prefer iterative methods, especially conjugate gradient (``CG'') methods, to
direct inversion methods, like singular-value decomposition or
orthogonalization by Givens rotations.

The advantages of conjugate-gradient methods are numerous. The matrix *A*^{T}*WA*
does not need to be computed and stored. The algorithm ``only'' requires the
computation of the gradient *g* = *A*^{T}*W*.*e* (*e*: residuals) and of *WA*.*h*
(*h*: conjugate direction).
The vectors *e* and *h* are updated at each iteration of the CG algorithm.
*W* being diagonal, the product between *W* and a vector is trivial. Moreover,
if the matrix *A* is sparse, its product by a vector can also be easily
computed. If the problem is ill-conditioned, limiting the number of iterations
is supposed to have an effect similar to that produced by adding a damping
factor.

Finally, the convergence of the CG algorithm is controlled by three factors:

- the condition number of the problem;
- the choice of an initial estimate
*x*of^{0}*x*; the problem is modified into the resolution of: = 0. - the structure of the eigenspectrum of
*A*^{T}*WA*itself: the more gathered the eigenvalues are, the quicker the algorithm will converge.

I will use the conjugate-gradient method in the IRLS algorithm applied to
deconvolution. However, as I will show, its efficiency will not be optimal,
unless some precautions are taken. For what follows, I mainly consider the
case of the *L ^{1}* deconvolution, with

1/13/1998