Example 5.  Use the Broyden's method to solve the nonlinear system  

            

Solution 5.

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 Broyden's method to find a numerical approximation to the solution near  .  

Use the subroutine to get the answer.

Compare with the Newton-Raphson method.  

(ii)  Use the Broyden's method to find a numerical approximation to the solution near  .  

Compare with the Newton-Raphson method.  

(c) John H. Mathews 2005