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The Autofit procedure minimizes the sum of the squares of m non-linear functions (number of
layers) in n variables (number of elements, m ≤ n) by a modification of the LevenbergMarquardt algorithm [16]. The Jacobian matrix is calculated by a forward-difference
approximation. The procedure is called repeatedly with decreasing tolerance limits until the
best fit is achieved. Unless the model system is converging very slowly, this method supplies
the optimal fit in one step.
If the convergence is slow, Autofit may stop due to reaching the allowed maximum number of
iterations without obtaining a solution to the accuracy specified. Try to invoke Autofit some
more times. The slow convergence may indicate that the model does not describe properly the
examined system.
When the Modified tube layer variant of the Layers-on-Nanotube model is used, the above
iterations are nested into an external cycle, which iterates the wall thickness of nanotube. As
this model can be applied only for one layer, the convergence is usually not slower than in
simple cases.
There are some other characteristics to assist defining more complex models. If there is some
„a priori‟ knowledge on the relation of the various layers, thickness of the different layers may
be declared to be equal or can be kept constant. Elements or layers can be excluded from the
automatic fit calculation, if necessary.
When a set of layers is repeated in sequence on the sample, it is enough to define one
repeating unit and to set the number of the repetitions. E.g., two layers are defined in the
Model window and the Multilayers attribute is set to 3 in the Model tab of the Parameters
window, then a six-layer system is calculated, as illustrated in the next figure.
Defined
Calculated
The fitting routine is not restricted by the physical meaning of the parameters. When any of
the results seems to be unrealistic (e.g., a negative or too large layer thickness) a warning
message is displayed.
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