Solving ODEs and DDEs with residual control

We first consider the numerical integration of ordinary differential equations (ODEs) with Runge-Kutta methods that have continuous extensions. For some methods of this kind we develop robust and inexpensive estimates of both the local error and the size of the residual. We then develop an effective program, ddesd, to solve delay differential equations (DDEs) with time- and state-dependent delays. To get reliable results for these difficult problems, the code estimates and controls the size of the residual. The user interface of ddesd makes it easy to formulate and solve DDEs, even those with complications like event location and restarts.