Variable-step variable-order algorithm for the numerical solution of neutral functional differential equations

The variable-step variable-order algorithm based on predictor-corrector methods for neutral functional differential equations is described. The algorithm is implemented in interpolation mode, i.e. we use fixed-step formulation of Adams-Bashford Adams-Moulton formulas while all back-values are computed by interpolation. The local discretization error is estimated by Milne's device. Although the assumptions under which this estimate is asymptotically correct are rather strong and not always expected to be satisfied in practice, the numerical experiments up to date indicate that this algorithm can be quite accurate and efficient.