Newton's method for nonlinear systems

A system of nonlinear equations is expressed in the form F(x)=0, where F is a vector-valued function of the vector variable x: $F:{\bf {\rm R}}^n\rightarrow{\bf {\rm R}}^n$. Given an estimate x^(k) of a solution x^*, Newton's method computes the (hopefully improved) estimate x^(k+1) by setting the local linear approximation to F at x^(k) to zero and solving for x:

$$\begin{eqnarray*}F(x^{(k)})+J(x-x^{(k)})=0&\Rightarrow&J(x-x^{(k)})=-F(x^{(k)})\... ...)}=-J^{-1}F(x^{(k)})\\ &\Rightarrow&x=x^{(k)}-J^{-1}F(x^{(k)}). \end{eqnarray*}$$

In this calculation, J=J(x^(k)) is the Jacobian matrix of F at x^(k). Therefore x^(k+1) is defined by the formula

$$x^{(k+1)}=x^{(k)}-J^{-1}F(x^{(k)}),\ k=0,1,2,\ldots.$$ (1)

If J happens to be singular, then the Newton step is undefined, and a robust algorithm must be prepared to deal with such a situation. I will deal with this issue later. For now, I will simply assume that J(x^*)is nonsingular, in which case the continuity of J will ensure that J(x^(k)) is nonsingular for any x^(k) sufficiently near x^*.