There are many mathematical situations (such as optimization of functions in calculus and applications involving certain differential equations) that require the finding of the roots of an equation. But finding roots is not easy. For instance, for a polynomial equation of degree five or higher there is no general formula which gives the roots. In this case and in other more complicated equations, one tries to approximate the roots. The Newton-Raphson method is an iterative method for generating a sequence of approximations to a solution of a given equation. It is hoped that this sequence of approximations ``converges" to the root. That is, as the process is carried on, the approximations get as close as desired to the root. As is demonstrated in the exercises below, this does not always happen. Also, Newton-Raphson may lead to different roots if different starting guesses are used. This is explored in the first exercise.
The Newton-Raphson formula is
For further details of the method, see the text.