Evaluation of Functions and Jacobians

Performing iteration (7.2.3) requires the evaluation of Fi and Ji = (f, for i = 1,2. From the definition of Fi it can be seen that the solutions of the ODE (7.1.6) have to be determined and evaluated at the rit time points ti. Additionally, the derivatives of these solutions with respect to the parameters s = sq,6 ) are required. These derivatives are 

The computation of sensitivity matrices is the most time consuming part of the algorithm and the way how these matrices are computed has a large impact on the overall performance of the method. 

One way to compute sensitivity matrices is to integrate the variational differential equations. 

By rewriting these equations as ODEs we obtain the variational differential equations (VDE) 

The variational differential equations are two matrix differential equations corresponding to a system of nx nx H- ne) differential equations. The right hand side of these equations depends on evaluated along the trajectory x(t s). If 

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