22. Bisection root-finding

Root-finding algorithms find values (y) for which an equation y = F(x) equals zero. For example, the equation y = x² – 4 has roots at x = 2 and x = -2 because at those values y = 0.

To find roots using binary subdivision, the program starts with an interval, x₀ ≤ x ≤ x₁, which we suspect contains a root. For the method to work, F(x₀) and F(x₁) must have different signs. In other words, one must be greater than zero and one must be less than zero.

To look for the root, the method picks the middle X value, x_new = (x₀ + x₁) / 2, and then calculates F(x_new). If the result is greater than zero, then you know that the root lies between x_new and whichever of x₀ and x₁ gives a value less than zero.

Similarly, if the result is less than zero, you know that the root lies between x_new and whichever of x0 and x1, whichever gives a value greater than zero.

Now, you update x₀ and x₁ to bound the new interval. You then repeat the process until F(x₀) and F(x₁) are within some maximum allowed error value.

Write a program that uses binary subdivision to find roots for equations. Make the equation a delegate parameter to the main method so you can easily pass the method different equations. Use the program to find the roots for the following equations:

For extra credit, make the program draw the equations and their roots.