# The Solution of Systems of Nonlinear Algebraic Equations

• Siegfried Selberherr
Chapter

## Abstract

The main result obtained in the preceding chapter is that discretization of the basic semiconductor equations yields a large system of nonlinear algebraic equations with the values of the dependent variables of the differential equations at discrete points as unknowns. For the considerations in this chapter we adopt the following nomenclature for the system of discretized equations:
$$\vec F(\vec w) = 0$$
(7-1)
$$\vec F = \left( \begin{gathered} \vec f_\psi (\vec w) \hfill \\ \vec f_n (\vec w) \hfill \\ \vec f_n (\vec w) \hfill \\ \end{gathered} \right)$$
(7-2)
$$\vec w = \left( \begin{gathered} {\vec \psi } \hfill \\ {\vec n} \hfill \\ {\vec p} \hfill \\ \end{gathered} \right)$$
(7-3)
$${\vec F}$$ is a vector funetion of rank three which itself consists of the vector funetions $$\vec f_\psi \vec f_n \,and\,\vec f_p$$. These vector funetions correspond to the discrete approximations for the Poisson equation and the continuity equations, respectively. The vector of unknowns $${\vec w}$$ is also comprised by three vectors which are formed by the values of the electrostatic potential $${\vec \psi }$$, electron concentration $${\vec n}$$ and hole concentration $${\vec p}$$ at discrete points of the Simulation geometry. We shall assume that the rank of all three vector funetions $$\vec f_\psi \vec f_n ,\vec f_p$$ and the three vectors $$\vec \psi ,\vec n,\vec p$$ equals n. This is not a necessary assumption but it will simplify the notation. It may well happen for practical applications that the rank of $${\vec \psi }$$ differs from the rank of $${\vec n}$$ and $${\vec p}$$ (e.g., when the Laplaee equation is solved in an insulator). For our purpose, the scalar rank of $${\vec F}$$ and $${\vec w}$$ is 3·n.

## Keywords

Iterative Method Jacobian Matrix Initial Guess Newton Method Convergence Property
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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