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

Section 8.1.4.5. Recommendations

E. Princea and P. T. Boggsb

a NIST Center for Neutron Research, National Institute of Standards and Technology, Gaithersburg, MD 20899, USA, and bScientific Computing Division, National Institute of Standards and Technology, Gaithersburg, MD 20899, USA

8.1.4.5. Recommendations

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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:

  • (1) In cases where the fit is poor, owing to an incomplete model or in the initial stages of refinement, methods based on the quadratic approximation to S (quasi-Newton methods) often perform better. This is particularly important when the model is close to a more symmetric configuration. These methods are more expensive per iteration and generally require more storage, but their greater stability in such problems usually justifies the cost.

  • (2) With small residual problems, where the model is complete and close to the solution, a safeguarded Gauss–Newton method is preferred. The trust-region implementation (Levenberg–Marquardt algorithm) has been very successful in practice.

  • (3) The best advice is to pick a good implementation of either method and stay with it.








































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