Exercise 4.  Observe that the subroutine PseudoNewtonSystem involves vector functions and is not dependent on the dimension.
Use the subroutine NewtonSystem to solve the nonlinear system in 3D space:  

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

Solution 4.

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

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

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

Compare the Pseudo-Newton result with Newton-Raphson's result.

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

Compare the Pseudo-Newton result with Newton-Raphson's result.

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

Compare the Pseudo-Newton result with Newton-Raphson's result.

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

Compare the Pseudo-Newton result with Newton-Raphson's result.

We are done.

Aside.  We can have Mathematica solve the system analytically.  There is a surprise.

Since Mathematica performs its solution using complex number arithmetic, the first four solutions are extraneous.
The solutions that we seek are the latter four solutions where x, y, and z are real numbers.  

(c) John H. Mathews 2005