A new class of methods for solving nonlinear equations (Q1334489)

Writing the nonlinear problem \(F(x)= 0\) with \(F: D\subset \mathbb{R}^ n\to \mathbb{R}^ n\) as a differential equation \(\partial x/\partial t=- F(x)^{- 1} F(x)\) along the Newton homotopy and choosing the initial value \(x(0)= x^ k\) as the \(k\)th approximation of the solution \(x^*\), the authors derive a class of implicit iterative methods for solving nonlinear equations via Runge-Kutta methods for initial value problems of differential equations. Convergence and implementations of the methods are discussed and illustrated with numerical examples.