next up previous
Next: An example of the Up: Newton's method Previous: Introduction

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

 \begin{displaymath}
x^{(k+1)}=x^{(k)}-J^{-1}F(x^{(k)}),\ k=0,1,2,\ldots.
\end{displaymath} (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*.



 

Mark S. Gockenbach
2003-01-23