Solvability of Nonlinear Equations

To distinguish this case from the case with a stationary reference system, the solution is sometimes called a quasistatic equilibrium. 

Evaluating F defined by (3.1.1) for an arbitrary point results in a residual = The first rip components of bo have the physical dimension of a force. 

This force must be added to the system to transform x into an equilibrium. The goal of a numerical scheme is to iteratively reduce this force to zero by incrementally changing x. 

In the sequel we will discuss the numerical solution of a set of nonlinear equations (3.1.1) by numerical methods such as Newton s method. Newton s method is the first method to be used if one has to solve nonlinear equations. There are many extensions of this method. We will focus on path following methods for one-parametric problems of type (3.1.2). 

The following theorem states conditions for the solvability of a system of nonlinear equations  

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