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
R_{1}(*θ*),
R_{2}(*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
*σ*_{NOE}(*θ*) and the auto relaxation rate
R_{1}(*θ*).
Taking equation (7.6) below the transformed relaxation equations are

whereas the relaxation equations are the
R_{1}(*θ*),
R_{2}(*θ*),
*σ*_{NOE}(*θ*).

The relax user manual (PDF), created 2020-08-26.