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 ).
Figure: The Preconditioned Conjugate Gradient Squared Method