International
Tables for Crystallography Volume C Mathematical, physical and chemical tables Edited by E. Prince © International Union of Crystallography 2006 
International Tables for Crystallography (2006). Vol. C, ch. 8.1, p. 685

One situation in which the Gauss–Newton algorithm behaves particularly poorly is in the vicinity of a saddle point in parameter space, where the true Hessian matrix is not positive definite. This occurs in structure refinement where a symmetric model is refined to convergence and then is replaced by a less symmetric model. The hypersurface of S will have negative curvature in a finite sized region of the parameter space for the lesssymmetric model, and it is essential to use a safeguarded algorithm, one that incorporates a line search or a trust region, in order to get out of that region.
On the basis of this discussion, we can draw the following conclusions:
