Example 4.  Show how Newton's method is in reality a fixed point iteration scheme.  Use the system    
            

Solution 4.

First, enter the coordinate functions and construct the vector function using Mathematica, and then find the Jacobian matrix .  

Construct the Newton-Raphson fixed point function .

Use fixed point iteration and the above fixed point function to find a numerical approximation to the solution near .  

A fixed point satisfies the equation  .  Our last approximation is stored in  ,  check it out.  

Accuracy is determined by the tolerance and number of iterations.  How accurate was the solution?

Compare these values with those obtained with the Newton-Raphson method for finding a numerical approximation to the solution near .  

A solution to the system satisfies .  Our last approximation is stored in , check it out.  

Accuracy is determined by the tolerance and number of iterations.  How accurate was the solution?

This Newton-Raphson solution is the same as the one obtained using fixed point iteration.

Use fixed point iteration and the above fixed point function to find a numerical approximation to the solution near .  

A fixed point satisfies the equation  .  Our last approximation is stored in  ,  check it out.  

Compare these values with those obtained with the Newton-Raphson method for finding a numerical approximation to the solution near .  

A solution to the system satisfies .  Our last approximation is stored in , check it out.  

This Newton-Raphson solution is the same as the one obtained using fixed point iteration.

(c) John H. Mathews 2004