Newton-Raphson Method

Newton-Raphson Method

Take a vector-valued function , and assume that we want to find a root of , that is, we look for such that

The Newton-Raphson method is an iterative method to find the root of . We start from an initial guess , and we iteratively update the guess by using the following formula:

where is the derivative matrix of at , and is the inverse of the derivative matrix.

If we set , then we can rewrite the update formula as:

which can be solved for , and then we can update .

Convergence

There are a few conditions for which the Newton-Raphson method converges to the root .

1. The derivative has an inverse at the root, i.e., is invertible.
2. The initial guess is sufficiently close to the root .
3. The function is sufficiently smooth close to the root .

At each -th iteration, the error from the real solution is given by:

Around the root ,

and

which gives