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Subsections
2.6 Newton Methods
The discretization of the nonlinear PDE systems leads to a large system of nonlinear algebraic
equations. There are various methods which solve such systems. Among them the most popular is a Newton method with it's variations.
Generally we have a system of N nonlinear algebraic equations

(2.32) 
assuming that the functions have continues first order derivations with respect to .
If the values
of the th approximate solution are known, we write Taylor series expansion of the functions in the vicinity of ,

(2.33) 
for
, where
.
By choosing for the exact value and neglecting the terms of order
we have from (2.33)

(2.34) 
The matrix

(2.35) 
is called Jacobi matrix. In the case when this matrix is nonsingular the linear equation system (2.34) can be solved. Depending on the problem itself different types of linear solvers should be applied [11].
The solutions of the linear system (2.34) are the increments
and with these increments we can calculate new approximation,

(2.36) 
The determination of damping value is the question of trial and error. The sequences often used for subsequent trials are,
or 
(2.37) 
is accepted if the following criterion is fulfilled,

(2.38) 
where we use Euclidian norm.
Since the calculation of the Jacobi matrix is a very computer resourcesconsuming procedure, one can for a number of subsequent Newton iterations also use the same Jacobi matrix (2.35). In that case on obtains,

(2.39) 
The simplified Newton method obviously has a worse convergence behavior.
2.6.4 Damping
In the cases where the convergence of the Newton scheme is attainable only for a very small time steps, methods for the enforcing of the convergence must be applied.
The method which has also successfully been applied in semiconductor device modelling [4], is the socalled BankRose algorithm [12].
Figure 2.3:
BankRose algorithm.

The algorithm is represented in Figure 2.3. It controls the behavior of Newton scheme through a damping factor .
The parameter of crucial importance for the algorithm is . In Figure 2.3 we take as initial estimate for . Alternatively, we have used
, where is the counter of failed tests
.
The algorithm searchs through the vincinity of the initial solution
for the shortest path through the attraction region of the Newton scheme to reach the next approximation
. In the cases when the BankRose algorithm fails the parameter grows infinitely and an improved approximation
is never found.
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H. Ceric: Numerical Techniques in Modern TCAD