Exercise 5.  Observe that the subroutine NewtonSystem 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 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.  

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

A solution to the system satisfies .  Our last approximation is stored in , check it out.  

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

A solution to the system satisfies .  Our last approximation is stored in , check it out.  

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

A solution to the system satisfies .  Our last approximation is stored in , check it out.  

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

A solution to the system satisfies .  Our last approximation is stored in , check it out.  

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 2004