Runge-Kutta methods and inverse Hermite interpolation

For an initial value problem relative to the first order ordinary differential equation and a given value, one desires to compute an abscissa, such that the value of the solution at that point equates the desired value. Our solution is a combination of dense output for Runge-Kutta methods and inverse Hermite interpolation. We are interested to save function evaluations in inverse interpolation. A MATLAB implementation is presented and some numerical example are given. The MATLAB approach allow us to consider a vector of desired values, and to divide the range of solution values into equal subintervals and to find the corresponding abscissas.