= Multivariable Newton iteration
{title2=$DG(x_k)s_k=-G(x_k),\quad x_{k+1}=x_k+s_k$}
= Newton's method for nonlinear systems
{c}
{synonym}
The finite-dimensional <Newton root-finding iteration> for a vector equation $G(x)=0$. At a point with invertible <Jacobian matrix>, solve the displayed <linear system>. For $G=\nabla F$, its <Jacobian matrix> is the <Hessian matrix>, giving <Newton's method in optimization>. The update solves the affine first-order approximation $G(x_k)+DG(x_k)s_k=0$; it can converge to any nondegenerate <critical point>, including a <saddle point>.
Back to article page
