Iterative Solutions of Nonlinear Equations Using Excel Solver: Interactive Simulation

This simulation was prepared using Mathematica. Download the free Wolfram player, and then download the simulation CDF file (link given below or click on figure to download). Try to predict the behavior when a parameter changes before using a slider to change that parameter.

prepared by: Heikki Ruskeepää; open content licensed under CC BY-NC-SA, view source code

Instructions:
Drag the locator to see how the iterations proceed when solving a set of two nonlinear equations using Newton’s method. The red and blue curves represent the points in the plane where the two equations hold. The two points of intersection of the red and blue curves are the solutions of the nonlinear system. The small light-blue points are initial points that lead to convergence to the point in the first quadrant. From the white parts of the plot, the convergence is to the point in the third quadrant.
Try to answer these questions before determining the answer with the simulation. We suggest that you write down the reasons for your answers.
  1. Does picking an initial guess in the first quadrant converge to the solution in the first or third quadrant?
  2. Could initial guesses that are close to each other converge to different solutions?