A TCAD task definition begins with a set of independent variables and/or parameters, a set of dependent variables, and the existing or simulated functional relationship between the two sets. Throughout this thesis, this functional relationship is referred to as a TCAD model. A TCAD model can have various level of complexity and is inherently nonlinear in nature. In general its evaluation consists of the execution of some underlying simulation tools that involves highly complex solution techniques which are computationally expensive. Furthermore, they are not suited for analytical manipulation and the use of numerical approximation is normally required. Typically, they have a continuous and well-behaved functional behavior. These characteristics influence the choice of the appropriate solution algorithm for any particular task.
It is clearly desirable to have a standard model formulation suitable for the description of the diverse tasks algorithms. In mathematical terms such an expression takes the form:
where
are vectors of 
model outputs, 
independent variables and 
model
parameters respectively. 
can also be formulated in terms of the
individual output functions as:
where each
represent the model evaluation that relates the input and parameter vectors to the i-th response.
In the rest of this chapter the different task algorithms are presented and discussed. They include: