Example 3.  Use the Pseudo-Newton's method to solve the nonlinear system  

             Solution 3.

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

Second, graph the curves 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 with the Newton-Raphson method.  

Observation.  The results for the subroutines NewtonSystem and PseudoNewtonSystem are very close.  Obviously, the Jacobian and its approximation agree to several significant digits.

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

Compare with the Newton-Raphson method.  

Observation.  The results for the subroutines NewtonSystem and PseudoNewtonSystem are very close.  Obviously, the Jacobian and its approximation agree to several significant digits.

(c) John H. Mathews 2005