PHZ4151C/5156C - Solution to Project 7

Solving Nonlinear Equations

Use the Bisection Method to find the root of a function describing the equilibrium angle from the horizontal of a pair of equal springs suspending a mass.

Example:
> ./BisectionMethod

Output:

 1        0.321963
 2     -0.00834326
 3        0.104785
 4       0.0385497
 5       0.0129939
 6      0.00183198
 7     -0.00337497
 8    -0.000801823
 9     0.000507435
10    -0.000149097

f(0.53252) = -0.000149097
The value of theta in degrees is 30.5111

Source Code

False Position Method

Example:
> ./FalsePositionMethod

Output:

  1      -0.0791721
  2      -0.0790662
  3      -0.0789577
  4      -0.0788467
...      ...
704      -0.000175035
705      -0.000173194
706      -0.000171373
707      -0.000169571

f(0.532477) = -0.000169571
The value of theta in degrees is 30.5087

Source Code

Newton-Raphson Method

Example:
> ./NewtonRaphsonMethod

Output:

 1       0.0524164
 2      0.00714035
 3      0.00020607
 4     1.87727e-07

f(0.532827) = 1.87727e-07
The value of theta in degrees is 30.5287

Source Code

Shawn Havery (PHZ4151C in Spring 2008):

The results of the successive approximations for the three methods are plotted here. Note that a log scale has been used for the iterations due to the large disparity between the rates of convergence of the False Position Method versus the other two.

The three methods all obtained the same root with almost similar precision:

  30.5111 degrees (Bisection Method)
  30.5087 degrees (False Position Method)
  30.5287 degrees (Newton-Raphson Method)

This is not surprising since the desired precision of f(theta) = 0 ± 0.00017 was hard-coded into every program. The three methods did not, however, have the same rate of convergence to the answer. In this particular case at least, the Newton-Raphson Method converged the fastest (4 iterations), followed by the Bisection Method (10 iterations). The slowest method by far was the False Position Method, which took a whopping 707 iterations to converge on the root within the desired precision. This result was unexpected, as the False Position Method uses more information about the function than does the Bisection Method, and thus it was expected to perform better. The Newton-Raphson Method, however, was indeed the fastest, which was expected given the sophistication of that particular method.