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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 [188]).