Nonlinear finite elements/Newton Raphson method

The Newton-Raphson methodEdit

The Newton-Raphson method is a technique used to find the roots of nonlinear algebraic equations. The method is also called Newton's method.

Let us revisit Newton's method of finding roots in the context of an equation with one degree of freedom. Let the nonlinear equation be

 

We wish to find one root of the equation (recall that if   is linear or quadratic, we can find the root in exact form).

Let us select a point   on the curve (see Figure 1). The slope of the curve at point   is  . Therefore, the equation of the tangent to the curve at   is

 

At  , the tangent meets the  -axis at  . Therefore we have

 

Recall that the root of   is the point where the curve of the function meets the  -axis. Therefore, we can repeat the process until we arrive at the root. The Newton iteration scheme is given by

 
 
Figure 1. Newton's method of finding roots.

The iterative procedure is terminated when the following condition is met:

 

where   is a small value that depends on the desired accuracy and on machine precision. The Newton-Raphson method gives a quadratic convergence rate. That is, the difference between the answer and the approximate solution is proportional to the previous difference squared. This can be seen by Taylor expanding the non-linear equation and its derivative around the root, and inserting this into the expression for  .