Shape-Preserving Local Interpolation for Plotting Solutions of ODEs

We investigate the use of piecewise rational interpolants ofDelbourgo and Gregory in an important and widely occurring application.We propose the following algorithm for visually pleasing plotsof the solution of an ordinary differential equation (ODE):use piecewise cubic Hermite interpolation where it can be shownto preserve shape (monotonicity and/or convexity) and also wherethere is no shape to preserve, otherwise use the appropriateconvex or monotone piecewise rational interpolant. Bounds arederived which enable efficient plotting of the rational interpolants.This scheme should be useful in any context where both solutionand derivative of a function are available as data.