Definition:Newton-Raphson Method

Definition

Let $f: \R \to \R$ be a real function which has a root which is to be found.

The Newton-Raphson method is an iterative technique for achieving this, by setting up the following iterative function:

$x_{n + 1} = x_n - \dfrac {\map f {x_n} } {\map {f'} {x_n} }$

where $x_0$ is a first approximation to the root.

Convergence is not guaranteed.

For each of $i \in \set {1, 2, \ldots, n}$, Let $f_i: \R^n \to \R$ be a real-valued function which has a set of roots to be found.

The Newton-Raphson method is an iterative technique for achieving this, by setting up the following iterative function:

$\mathbf x_{m + 1} = \mathbf x_m - \inv {\mathbf J} {\mathbf x_m} \map f {\mathbf x_m}$

where:

$\mathbf x_m$ and $\map f {\mathbf x_m}$ are column matrices of order $n$

$\mathbf x_0$ is the column matrix of the initial first approximation to the set of roots

$\inv {\mathbf J} {\mathbf x^n}$ is the inverse of the Jacobian evaluated at $\mathbf x_m$.

This entry was named for Isaac Newton and Joseph Raphson.