Results

The following graph displays the results visually:

Employing the three C++ programs, I was able to analyze the provided data with Forward Euler’s, Backward Euler’s, and the Runge-Kutta methods. Using Microsoft Excel to plot the various sets of data versus the actual test data, we can graphically see how well the methods approximate the curvature of the eye.

The Forward Euler’s method (using 21 points) has an average error of -0.044868%. This translates into an actual average error of 0.0015205 millimeters. Backward Euler’s method (using 21 points) has an average error of 0.026055%, translating into an actual average error of 0.0013263 millimeters. The Runge-Kutta (using 21 points) method has an average error of -0.0027841%, translating into an actual average error of 0.00010728 millimeters. When 51 points are used for the Runge-Kutta evaluation, the average error drops to -0.0010919% and the actual average error drops to 0.000091948 millimeters.