As was noted before, Gauss-Newton minimization may fail where the new parameter values once the \(\delta _k\) values have been applied, after a minimization cycle, produce a higher \(\chi ^2\) value. This indicates that correlation has basically corrupted the convariance matrix, despite use of SVD. When this occurs yet another trick, called automatic Marquardt damping is brought into play.
Marquardt damping was first developed by Kenneth Levenberg and then rediscovered by Donald Marquardt so it is commonly called Levenberg-Marquardt minimization. In this, an ad hoc term, known as the Marquardt damping factor, is used that increases the importance of the diagonal Hessian terms and down-weights the off-diagonal terms. This decreases the impact of correlation. The minimizer normally used in GSAS-II will increase this Marquardt damping factor after any set of shifts are computed that raise \(\chi ^2\) and the shifts are recomputed. This is repeated until the \(\chi ^2\) value drops.