ABSTRACT
This chapter illustrates the construction of a dense output formula by considering an extension to the well-known optimal Runge–Kutta (RK2) process. Dense output is desired, implying a large number of intermediate points, the extra computational cost could be considerable. Fortunately, this unpleasant prospect can be avoided by forming an interpolant or dense output formula, sometimes termed a continuous extension to the RK formula. The construction of dense output formulae usually demands additional stages to provide necessary parameters. A third order case serves to demonstrate this feature. Thus the derivative of the dense output solution is continuous. This property is a necessary one for the first same as last model chosen but, in other cases where the solution of the equations of condition is non-unique, it need not be satisfied. The extension of the variable step RK program, to include dense output is very straightforward. A further important application of the continuous extension is the solution of the inverse interpolation problem.
