The chi-squared equation is itself dependent on the relaxation equations through the back-calculated relaxation data R(θ). Letting the relaxation values of the set R(θ) be the R1(θ), R2(θ), and NOE(θ) an additional layer of abstraction can be used to simplify the calculation of the gradients and Hessians. This involves decomposing the NOE equation into the cross relaxation rate constant σNOE(θ) and the auto relaxation rate R1(θ). Taking equation (7.6) below the transformed relaxation equations are
whereas the relaxation equations are the R1(θ), R2(θ), σNOE(θ).