Newton's Method

Newton's method can be generalized to a system of nonlinear equations by expanding ${\bf f}$in a Taylor series around ${\bf x} = {\bf \alpha}$ and keeping only the linear term. The result is  

$${\bf J} \left(x^{\left(n \right)} \right) \cdot \left( x^{\... ...ight)} \right) = - {\bf f} \left(x^{\left(n \right)} \right).$$ (250)

Here ${\bf J}$ is the Jacobian defined as ${\bf J} \equiv \left[ \frac{\partial f_i}{\partial x_j} \right]$.Thus, Newtons method involves a linearization of the nonlinear system of equations. To assure convergence, it requires an initial guess which is close enough to the desired solution, and a step size which is small enough. These requirements can be fulfilled by a systematic procedure called parametric continuation. IMSL subroutines based on Newton's method, like NEQNF and NEQNJ, can be used to solve in practice nonlinear systems of equations.