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 [10,21,32].

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 **n** [11,32].