The Newton - Raphson Method

The Newton – Raphson method offers a useful method of finding the roots of functions.

We start from the Taylor series for a function

The first two terms on the right give us the equation of the tangent at

This tangent line intersects theaxis atso

Rearrangement of this givesand we can generalise this to giveIteration gives better approximations to the rootas shown below.

The method is not infallible. The method may not converge to the desired root or diverge from a required root, especially ifis not chosen properly or the gradient is changing rapidly, or if one of theis such thatis not defined eg when finding the root close to zero of a function with a square root or log.