In BiCG, the residual vector 
can be regarded as
the product
of 
and an 
th degree polynomial in 
, that is
This same polynomial satisfies
so that
This suggests that if 
reduces to a smaller
vector , then it might be advantageous to apply this
``contraction'' operator twice, and compute 
.
Equation (
) shows that the iteration coefficients can
still be recovered from these vectors, and it turns out to be easy to
find the corresponding approximations for 
. This
approach leads to the Conjugate Gradient Squared method (see
Sonneveld [192]).
Figure: The Preconditioned Conjugate Gradient Squared Method
