Gauss adaptive methods

Based on the above, we can develop an "adaptive" Gauss-Newton method for parameter estimation with equality constraints whereby the set of active constraints (which are all equalities) is updated at each iteration. An example is provided in Chapter 14 where we examine the estimation of binary interactions parameters in cubic equations of state subject to predicting the correct phase behavior (i.e., avoiding erroneous two-phase split predictions under certain conditions). 

The adaptive methods that use the Gauss-Lobatto algorithms have the following pros and cons. 

Figure 3.61. Relation between the order parameters S ) and (S >. The physical boundaries of (S ) and (S >. which follow from [3.7.13 and 14], are indicated by solid curves. The dashed curve indicates combinations of the two order parameters which correspond to Gauss type distributions. The insets show the shapes of the distribution function N 6) for 9 between 0° and 90° as calculated using the maximum entropy method. (Adapted from M.A. Bos and J.M. Kleijn, Biophys. J. 68 (1995) 2566.)...

When the equations are nonlinear in the parameters, the parameter estimates are obtained by minimizing the objective function by methods like that of Newton-Raphson or that of Newton-Gauss or an adaptation of the latter such as the Marquardt algorithm [1963], In the latter case parameters are iteratively improved by the following formula ... 

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