Exercise 6.  Observe that the subroutine Broyden involves vector functions and is not dependent on the dimension.
Use the subroutine Broyden to solve the nonlinear system in 3D space:  

           
          
          
    
Hint.  There are four solutions.  Good starting vectors are  .  

Solution 6.

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

Second, graph the surfaces  ,    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 Broyden's result with Newton-Raphson's result.

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

Compare Broyden's result with Newton-Raphson's result.

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

Compare Broyden's result with Newton-Raphson's result.

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

Compare Broyden's result with Newton-Raphson's result.

(c) John H. Mathews 2005