Vector and Matrix Variability Type in Sparse Matrix Algorithms

In the computational design of nonlinear dynamic systems described by ODE of the form $$\hat{f}\left( {x,\frac{d}{{dt}},t,p} \right) = 0$$ (1) there usually exist three nested iteration loops, namely, 1) the outermost, p-loop (i.e., parameter optimization), 2) the t-loop (i.e., time integration) and 3) the innermost, x-loop (i.e., Newton iteration).