Example 1.  Use Newton's method to solve the nonlinear system  

            

Solution 1.

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

 

Second, graph the curves    and    using Mathematica.  The points of intersection are the solutions we seek.  

(i)  Use the Newton-Raphson method to find a numerical approximation to the solution near  .  

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

Do you think that iteration produced the solution ? Why ?  

Compare with Mathematica's built in routine.

Whose answer is best, ours or Mathematica's ?  How can this be ?  Find out how to increase the number of iterations in Mathematica's subroutine.

(ii)  Use the Newton-Raphson method to find a numerical approximation to the solution near  .  

Do you think that iteration produced the solution ? Why ?  

Compare with Mathematica's built in routine.