Equivalence of QN–LS and BQN–LS for affine problems

Previously, we studied methods to solve the coupled system of non-linear equations F(g)=pF(g)=p and S(p)=gS(p)=g. In this paper we take a closer look at two of them, the Quasi-Newton method with Least Squares Jacobian (QN–LS) and the Block Quasi-Newton method with Least Squares Jacobian (BQN–LS). We show that both are algebraically equivalent if one of the operators (FF or SS) is affine. This implies that for this type of problem there is no reason to use BQN–LS, as the results will be the same but for a higher computational cost.