judge from the behaviour of $\sigma^2$ as the iteration proceeds whether a satisfactory inversion has been achieved. A low value of $\sigma^2$ obviously guarantees that a potential closely reproducing the input $S_l$ has been found. Because the IP method is implemented interactively, there is an opportunity to examine the potential for oscillatory features. These might well be spurious and result from over-fitting noisy data. In such a case, one can reduce the basis dimensionality or raise the SVD limit and this generally allows one to achieve a smooth potential, often with only a small increase in $\sigma^2$. One must bear in mind that genuine oscillatory features, corresponding to non-locality in an $L$-independent local potential or to $L$-dependence of the underlying potential, can be necessary to achieve a precise representation of $S_{lj}$ or $S^J_{l'l}$.

We stress here that, although the context of our discussion is the determination of a tensor interaction from the non-diagonal $S$-matrix elements describing the scattering of spin-1 projectiles, the range of application is much more general.

### Derivation of non-diagonal perturbation expression. {#derivation}

We now outline the derivation of Eq.\[integ\]. The derivation can be applied to the general coupled channel inversion from non-diagonal $S$-matrix to non-diagonal potential. 