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
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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 less-symmetric 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:
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