5. A comparison of the three methods

After the discussion of Newton's Method, you might wonder why someone would want to use one of the other methods at all. Well, that's because Newton has some drawbacks. First of all, you can't use Newton to find double roots reliable. This is because in the calculation of the next approximation, you divide by $f'(x_n)$. When $f'(x_n)$ goes to zero, you get a very bad conditioned calculation.

A more serious problem is that, athough Newton is very fast, it is not very stable. To see what that means, try the following example: You see the problem; Newton is fooled in thinking that the function has a root at the right of the y-axis, because the function has a local minimum there. But the other methods aren't fooled, because they can know for certain that the function has a root in their interval. In practice, Bisection and Regula Falsi are often used to ``stabilize'' Newton's Method.

There is another problem with Newton; you need the first derivative of your function to find the root. But in general, the function $f$ might be the result of some other numerical calculations, so you can't directly find the derivative. Instead, you might need to use a numerical method for differentiation to find $f'(x)$ in the first place! So although Newton uses less iterations, every individual iteration might take much longer. There are some ways to prevent this, for example, by keeping $f'(x)$ the same for a few iterations. Another variant uses two points, calculates the slope of the line joining those two pints, and uses this value as an approximation to $f'(x)$. This method is called the Secant Method.