SENSITIVITY ANALYSIS OF INITIAL-VALUE PROBLEMS WITH APPLICATION TO SHOOTING TECHNIQUES

The solution y(x, x0, y0) of an initial-value problem (IVP) y'(x) = f(x, y), y(x0) = y0 at a point x is a differentiable function of the initial value y0 provided partial derivative f/partial derivative y is continuous. The derivative partial derivative y(x, x0, y0)/partial derivative y0 of y, which reflects the sensitivity of the IVP, is most frequently computed by a finite difference approximation. In connection with shooting techniques, this sensitivity analysis is the dominant part of the overall computation time and it can be accelerated if sophisticated methods are used. Several different strategies are possible and all lead to very different algorithms. Some of the most promising ones were implemented and compared. It turns out that the computation time can be reduced to less than one fourth compared with a finite difference approximation of partial derivative y(x. x0, y0)/partial derivative y0.