On the Newton iteration in the application of collocation methods to implicit delay equations.

We consider stiffly accurate collocation methods based on Radau nodes for the integration of initial value problems for implicit delay differential equations of the form \begin{eqnarray} \begin{array}{rcl} M\, y'(t) &\! = \!& f\Bigl( t,y(t),y\bigl( \alpha_1 (t,y(t))\bigr), \ldots ,y\bigl( \alpha_p (t,y(t))\bigr)\Bigr), \end{array} \end{eqnarray} where $M$ is a constant matrix and $\alpha_i (t,y(t))$ ($i=1,\ldots,p$) denote the deviating arguments, which are assumed to satisfy the inequalities $\alpha_i (t,y(t)) \le t$ for all $i$. % In a recent paper \cite{GuHa3} we have described how collocation methods based on Radau nodes can be applied to solve problems of this type. The aim of this paper is that of explaining the difficulties arising when solving the Runge–Kutta equations using stepsizes larger than delays and to design techniques able to efficiently overcome them.