Performance of the CG method is generally very sensitive to roundoff in the computations that may destroy the mutual conjugacy property. The method was actually neglected for many years until it was realized that a preconditioning technique can be used to accelerate convergence significantly [32][21][10].
Preconditioning involves modification of the
target linear system 
through application of a
positive-definite preconditioner 
that is closely related
to 
.
The modified system can be written as
Essentially,
the new coefficient matrix is 
.
Preconditioning aims to produce a more clustered
eigenvalue structure for and/or lower condition number than
for to improve the relevant convergence ratio.
However, preconditioning also adds to the computational effort
by requiring that a linear system involving (namely 
) be solved at every step. Thus, it is essential for efficiency
of the method that be factored very rapidly in relation
to the original . This can be achieved, for example, if
is a sparse component of the dense . Whereas the solution of an
dense linear system requires order of 
operations,
the work for sparse systems can be as low as order 
[32][11].
The recurrence relations for the PCG method can be derived for
Algorithm 2.2
after substituting 
and
. New search vectors 
can be used to derive the iteration process, and then
the tilde modifiers dropped. The PCG method becomes the following
iterative process.
Note above that the system 
must be solved repeatedly
for 
and that the matrix/vector products 
are required
as before.
