Exercise 9.  Observe that the subroutine ImprovedBroyden involves vector functions and is not dependent on the dimension.
Use the subroutine ImprovedBroyden to solve the nonlinear system in 3D space:  

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

Solution 9.

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

Compare with the standard Broyden's method.

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

Compare with the standard Broyden's method.

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

Compare with the standard Broyden's method.

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

Compare with the standard Broyden's method.

(c) John H. Mathews 2005